Real Number System
1.1 Ordered field properties
The real numbers form a field with a compatible total order. Addition, multiplication, additive/multiplicative inverses and order laws hold.
1.2 Absolute value
Absolute value gives distance from zero.
\[|x| = \begin{cases} x, & x \ge 0 \\ -x, & x < 0 \end{cases}\]
1.3 Distance
The distance between x and y is the absolute difference. \[d(x,y)=|x-y|\]
1.4 Basic absolute-value equivalence
An absolute-value bound is equivalent to a two-sided inequality. \[|x|<\varepsilon \Longleftrightarrow -\varepsilon<x<\varepsilon\]
1.5 Triangle inequality
The absolute value of a sum is at most the sum of the absolute values. \[|x+y|\le|x|+|y|\]
1.6 Reverse triangle inequality
The difference of absolute values is bounded by the absolute difference. \[||x|-|y||\le|x-y|\]
1.7 Intervals
Intervals are the basic connected subsets of \(\mathbb{R}\). \[(a,b), [a,b], (a,b], [a,b)\]
1.8 Upper bound
u is an upper bound of S if every element of S is at most u. \[x\le{}u \: \forall{} x\in{}S\]
1.9 Lower bound
l is a lower bound of S if every element of S is at least l. \[l\le{}x \: \forall{} x\in{}S\]
1.10 Maximum and minimum
A maximum/minimum is an element of the set that is respectively \(\ge\)/\(\le\) every other element. \[M=max \: S \Longleftrightarrow M\in{}S \text{ and } x\le{}M \: \forall{}x\in{}S\]
1.11 Supremum
sup S is the least upper bound. \[s=sup \: S\]
1.12 Infimum
inf S is the greatest lower bound. \[m=inf \: S\]
1.13 Least-upper-bound property (The Completeness Axiom)
Every nonempty subset of \(\mathbb{R}\) bounded above has a supremum in \(\mathbb{R}\). \[S\ne\varnothing \text{ and } S \text{ bounded above }\Longrightarrow sup \:S\in\mathbb{R}\]
1.14 Greatest-lower-bound property
Every nonempty subset of \(\mathbb{R}\) bounded below has an infimum in \(\mathbb{R}\). \[S\ne\varnothing \text{ and } S \text{ bounded below } \Longrightarrow inf \: S\in\mathbb{R}\]
1.15 Archimedean property
The natural numbers are unbounded above. \[\forall{}x\in\mathbb{R}, \: \exists{}\:n\in\mathbb{N}: n>x\]
1.16 Archimedean reciprocal form
There are arbitrarily small positive reciprocals of natural numbers. \[\forall\varepsilon>0, \: \exists{}\: n\in\mathbb{N}: 1/n<\varepsilon\]
1.17 Density of rationals
Between any two distinct real numbers lies a rational number. \[a<b \Longrightarrow \exists{}\:q\in\mathbb{Q}: a<q<b\]
1.18 Density of irrationals
Between any two distinct real numbers lies an irrational number. \[a<b \Longrightarrow \exists{}\:x\in \mathbb{R} \setminus \mathbb{Q}: a<x<b\]
1.19 Nested interval principle
A nested sequence of non-empty closed bounded intervals has non-empty intersection. \[I_{1}\supseteq{}I_{2}\supseteq\cdots \Longrightarrow \bigcap{}I_{n}\ne\varnothing\]
1.20 Finite intersection principle for compact intervals
A family of closed intervals with the finite-intersection property has nonempty total intersection. \[\text{ Every finite intersection }\ne\varnothing \Longrightarrow \bigcap{}I\alpha\ne\varnothing\]
1.21 Countability of \(\mathbb{Q}\)
The rational numbers are countable. \[|\mathbb{Q}|=|\mathbb{N}|\]
1.22 Uncountability of \(\mathbb{R}\)
The real numbers are not countable. \[|\mathbb{R}|>|\mathbb{N}|\]
Sequences
2.1 Sequence
A sequence is a function from \(\mathbb{N}\) to \(\mathbb{R}\) (or another set). \[(a_n)_{n=1}^\infty\]
2.2 Convergence
A sequence converges to L if its terms eventually lie within every \(\varepsilon\)-neighborhood of L. \[a_{n}\to{}L \Longleftrightarrow \forall\varepsilon>0 \: \exists{}N: n\ge{}N \Longrightarrow |a_{n}-L|<\varepsilon\]
2.3 Uniqueness of limit
A convergent sequence has exactly one limit. \[a_{n}\to{}L_{1} \text{ and } a_{n}\to{}L_{2} \Longrightarrow L_{1}=L_{2}\]
2.4 Eventually bounded
Every convergent sequence is eventually bounded, and hence bounded. \[a_{n}\to{}L \Longrightarrow \exists{}\:\:M,N: n\ge{}N \Longrightarrow |a_{n}|\le{}M\]
2.5 Limit laws
Limits preserve sums, scalar multiples, products and quotients when the quotient denominator limit is nonzero. \[lim(a_{n}\pm{}b_{n})=a\pm{}b; \:\: lim(ca_{n})=cL; \:\:lim(a_{n}b_{n})=ab\]
2.6 Squeeze theorem
A sequence between two sequences with the same limit has that limit. \[a_{n}\le{}b_{n}\le{}c_{n}, a_{n},c_{n}\to{}L \Longrightarrow b_{n}\to{}L\]
2.7 Absolute-value criterion
Convergence can be tested using absolute distance from the proposed limit. \[a_{n}\to{}L \Longleftrightarrow |a_{n}-L|\to0\]
2.8 Subsequence
A subsequence is obtained by selecting terms with strictly increasing indices. \[a_{n_k}, n_{1}<n_{2}<\cdots\]
2.9 Subsequence theorem
Every subsequence of a convergent sequence converges to the same limit. \[a_{n}\to{}L \Longrightarrow a_{n_k}\to{}L\]
2.10 Divergence to infinity
A sequence tends to +\(\infty\) when it eventually exceeds every real bound. \[a_{n}\to\infty \Longleftrightarrow \forall{}M\in\mathbb{R} \:\: \exists{}N: n\ge{}N \Longrightarrow a_{n}>M\]
2.11 Monotone sequence
Increasing/decreasing means successive terms are ordered consistently. \[a_{n+1}\ge{}a_{n} \text{ or } a_{n+1}\le{}a_{n}\]
2.12 Monotone convergence theorem
A monotone bounded sequence converges. \[a_{n}\uparrow \text{ and bounded above }\Longrightarrow a_{n} \text{ converges.}\]
2.13 Bolzano-Weierstrass theorem
Every bounded sequence in \(\mathbb{R}\) has a convergent subsequence. \[(a_{n}) \text{ bounded } \Longrightarrow \exists{} \:\:a_{n_k}\to{}L\]
2.14 Cauchy sequence
Terms eventually become arbitrarily close to each other. \[\forall\varepsilon>0 \:\exists\:{}N: m,n\ge{}N \Longrightarrow |a_m-a_{n}|<\varepsilon\]
2.15 Cauchy criterion for \(\mathbb{R}\)
A real sequence converges iff it is Cauchy. \[a_{n} \text{ converges } \Longleftrightarrow (a_{n}) \text{ is Cauchy.}\]
2.16 limsup
The limit superior is the limiting upper envelope of the sequence. \[\limsup_{n \to \infty} a_n = \lim_{n \to \infty} \left( \sup_{k \ge n} a_k \right)\]
2.17 liminf
The limit inferior is the limiting lower envelope. \[\liminf_{n \to \infty} a_n = \lim_{n \to \infty} \left( \inf_{k \ge n} a_k \right)\]
2.18 Convergence via limsup and liminf
A bounded sequence converges exactly when its limsup and liminf agree. \[a_{n} \to L \iff \limsup_{n \to \infty} a_{n} = \liminf_{n \to \infty} a_{n} = L\]
2.19 Cluster point of a sequence
L is a cluster point if some subsequence converges to L. \[\exists{}\:n_{k}: a_{n_k}\to{}L\]
2.20 Every bounded sequence has a cluster point
This is the Bolzano-Weierstrass consequence in \(\mathbb{R}\). \[(a_{n}) \text{ bounded } \Longrightarrow \text{It has a cluster point.}\]
Infinite Series
3.1 Series and partial sums
A series converges when its sequence of partial sums converges. \[\sum_{n} \text{ converges } \Longleftrightarrow S_{n}=\sum_{k=1}^na_{k} \text{ converges.}\]
3.2 Necessary condition
The terms of a convergent series must tend to zero. \[\sum_{n} \text{ converges } \Longrightarrow a_{n}\to0\]
3.3 Cauchy criterion for series
A series converges iff its tails can be made arbitrarily small. \[\forall \: \varepsilon>0, \:\:\exists{} \:N: m>n\ge{}N \Longrightarrow \left|\sum_{k=n+1}^ma_{k}\right|<\varepsilon\]
3.4 Geometric series
The geometric series converges exactly when |r|<1. \[\sum_{n=0}^{\infty} a r^{n} = \frac{a}{1-r}, \quad |r| < 1\]
3.5 Harmonic series
The harmonic series diverges. \[\sum_{n=1}^\infty\frac{1}{n}=\infty\]
3.6 p-series
The p-series converges exactly for p>1. \[\sum_{n=1}^\infty \frac{1}{n^p} \text{ converges} \Longleftrightarrow p>1\]
3.7 Comparison test
A nonnegative series dominated term-by-term by a convergent series converges. \[0\le{}a_{n}\le{}b_{n}, \sum b_{n}<\infty \Longrightarrow \sum a_{n}<\infty\]
3.8 Comparison divergence form
A series dominating a divergent nonnegative series also diverges. \[0\le{}b_{n}\le{}a_{n}, \sum b_{n} \text{ diverges} \Longrightarrow \sum a_{n} \text{ diverges.}\]
3.9 Limit comparison test
Positive series with a finite positive term ratio limit have the same convergence behavior. \[\lim_{n \to \infty}\frac{a_{n}}{b_{n}}=L, 0<L<\infty\]
3.10 Ratio test
Ratio limit less than one gives absolute convergence; greater than one gives divergence. If L=1, the test is inconclusive. \[L=\lim_{n \to \infty}\left|\frac{a_{n+1}}{a_{n}}\right|; \: \:L<1 \Longrightarrow \text{absolute convergence.}; \: L>1 \Longrightarrow \text{div}.\]
3.11 Root test
Root limsup less than one gives absolute convergence; greater than one gives divergence. \[L=\limsup_{n\to\infty}\sqrt[n]{|a_n|};\quad L<1\Longrightarrow\text{absolute convergence};\quad L>1\Longrightarrow\text{divergence}.\]
3.12 Integral test
For positive decreasing f, the series and corresponding improper integral have the same convergence behavior. \[\sum_{n=1}^\infty f(n) \text{ converges } \Longleftrightarrow \int_{1}^\infty f(x)dx \text{ converges }.\]
3.13 Alternating-series test
A decreasing positive sequence tending to zero yields a convergent alternating series. \[a_{n}\downarrow0 \Longrightarrow \sum_{n=1}^\infty(-1)^{n}a_{n} \text{ converges.}\]
3.14 Alternating-series remainder
The absolute error is no greater than the first omitted term. \[|R_{n}|\le{}a_{n+1}\]
3.15 Absolute convergence
Absolute convergence implies ordinary convergence. \[\sum_{n=1}^\infty|a_{n}|<\infty \Longrightarrow \sum_{n=1}^\infty a_{n} \text{ converges.}\]
3.16 Conditional convergence
A series is conditionally convergent if it converges but does not converge absolutely. \[\sum_{n=1}^\infty a_{n} \text{ converges and } \sum_{n=1}^\infty |a_{n}| diverges\]
3.17 Rearrangement theorem
Absolutely convergent series retain their sum under arbitrary rearrangements. \[\sum_{n=1}^\infty|a_{n}|<\infty \Longrightarrow \sum_{n=1}^\infty\pi(n)=\sum_{n=1}^\infty a_{n}\]
3.18 Riemann rearrangement theorem
A conditionally convergent real series can be rearranged to converge to any prescribed real number, or diverge to \(\pm\)\(\infty\). \[\text{Conditional convergence } \Longrightarrow \text{ arbitrary rearranged sums possible.}\]
Limits of Functions
4.1 Function limit
The \(\varepsilon\)–\(\delta\) definition describes the behavior of f near a, excluding the value at a itself. \[\lim_{x\to a} f(x)=L \:\: \Longleftrightarrow \:\:\:\forall\varepsilon>0 \: \exists\:\delta>0: 0<|x-a|<\delta \Longrightarrow |f(x)-L|<\varepsilon\]
4.2 Uniqueness of function limit
A finite limit, if it exists, is unique. \[\lim_{x\to a} f(x)=L_{1}=L_{2}\]
4.3 One-sided limits
A two-sided limit exists exactly when both one-sided limits exist and agree. \[\lim_{x\to a} f=L \Longleftrightarrow \lim_{x\to a^{-}}f=\lim_{x\to a^{+}} f=L\]
4.4 Limit laws
Limits preserve algebraic operations under the usual hypotheses. \[lim(f\pm{}g)=L\pm{}M;\:\: lim(fg)=LM; \:\:lim(f/g)=L/M\]
4.5 Squeeze theorem
A function trapped between two functions with the same limit has that limit. \[g\le{}f\le{}h \text{ and } g,h\to{}L \Longrightarrow f\to{}L\]
4.6 Sequential criterion for limits
A function has limit L at a iff every sequence approaching a through the domain has images approaching L. \[x_{n}\to{}a, x_{n}\ne{}a \Longrightarrow f(x_{n})\to{}L\]
4.7 Infinite limits
The function tends to infinity if it eventually exceeds every bound near a. \[\lim_{x\to a} f(x)=\infty \Longleftrightarrow \forall{}M>0\:\: \exists \:\: \delta>0: 0<|x-a|<\delta \Longrightarrow f(x)>M\]
4.8 Limits at infinity
A finite limit at infinity describes eventual closeness as x becomes arbitrarily large. \[\lim_{x\to \infty} f(x)=L\]
4.9 Asymptote criterion
A horizontal asymptote y=L corresponds to a finite limit at \(\pm\)\(\infty\). \[\lim_{x\to \infty} f(x)=L \:\: \Longrightarrow \:\:y=\text{L is a horizontal asymptote.}\]
Continuity
5.1 Continuity at a point
Continuity means the function value agrees with the local limit. \[f \text{ continuous at a }\Longleftrightarrow \lim_{x \to a} f(x)=f(a)\]
5.2 Sequential continuity
Continuity is equivalent to preservation of convergent sequences. \[x_{n}\to{}a \Longrightarrow f(x_{n})\to{}f(a)\]
5.3 Algebra of continuous functions
Sums, products, quotients where defined, and compositions of continuous functions are continuous.
5.4 Intermediate Value Theorem
A continuous function on an interval assumes every value between any two of its values. \[f\in{}C[a,b], y \text{ between } f(a),f(b) \Longrightarrow \exists{} \:\:c: f(c)=y\]
5.5 Bolzano theorem
A continuous function with opposite signs at endpoints has a zero in the interval. \[f(a)f(b)<0 \Longrightarrow \exists{} \:\: c\in(a,b): f(c)=0\]
5.6 Extreme Value Theorem
A continuous function on a compact interval attains both its maximum and minimum. \[f\in{}C[a,b] \Longrightarrow \exists{}\:\:x_{1},x_{2}: f(x_{1})\le{}f(x)\le{}f(x_{2})\]
5.7 Uniform continuity
One \(\delta\) works for all pairs of points in the domain. \[\forall\varepsilon>0 \:\exists\:\delta>0: |x-y|<\delta \Longrightarrow |f(x)-f(y)|<\varepsilon\]
5.8 Heine-Cantor theorem
Continuity on a compact interval implies uniform continuity. \[f\in{}C([a,b]) \Longrightarrow f \text{ is uniformly continuous.}\]
5.9 Lipschitz condition
A global linear bound on changes implies uniform continuity. \[|f(x)-f(y)|\le{}M|x-y| \Longrightarrow f \text{ is uniformly continuous.}\]
5.10 Uniform continuity preserves Cauchy sequences
A uniformly continuous function maps Cauchy sequences to Cauchy sequences. \[(x_{n}) \text{ Cauchy } \Longrightarrow (f(x_{n})) \text{ Cauchy }\]
5.11 Types of discontinuity
Removable, jump and infinite discontinuities describe standard ways continuity can fail.
Differentiability
6.1 Derivative
The derivative is the limit of the difference quotient. \[f'(a)=\lim_{h \to 0} \frac{f(a+h)-f(a)}{h}\]
6.2 Differentiability implies continuity
Differentiability at a point implies continuity there. \[f'(a) \text{ exists } \Longrightarrow f \text{ is continuous at a.}\]
6.3 Geometric meaning
The derivative is the slope of the tangent line to the graph. \[slope=f'(a)\]
6.4 Derivative as linear approximation
For small h, the change in f is approximately f’(a)h. \[f(a+h)\approx{}f(a)+f'(a)h\]
6.5 Derivative rules
Constants, sums, products, quotients and compositions obey standard differentiation rules. \[(fg)'=f'g+fg'\]
6.6 Chain rule
The derivative of a composition is the product of the outer and inner derivatives. \[(f\circ g)'(x)=f'(g(x))g'(x)\]
6.7 Power rule
Derivative of \(x^n\). \[\frac{d}{dx}\left(x^{n}\right) = n x^{n-1}\]
6.8 Exponential rule
Derivative of \(e^x\). \[(e^x)'=e^x\]
6.9 Logarithm rule
Derivative of ln x. \[(ln \: x)'=1/x\]
6.10 Basic trigonometric derivatives
Standard trigonometric derivatives. \[(sin x)'=cos x; \:\: (cos x)'=-sin x; \:\:(tan x)'=sec^{2}x\]
6.11 Inverse-function derivative
For a differentiable inverse with nonzero derivative, the inverse derivative is reciprocal. \[\left(f^{-1}\right)'(y) = \frac{1}{f'\left(f^{-1}(y)\right)}\]
6.12 Rolle’s theorem
Equal endpoint values force a stationary point inside. \[f(a)=f(b) \Longrightarrow \exists{}\:\:c\in(a,b): f'(c)=0\]
6.13 Lagrange Mean Value Theorem
Some tangent slope equals the average slope across the interval. \[\exists{}\:\:c\in(a,b): f'(c)=\frac{f(b)-f(a)}{b-a}\]
6.14 Cauchy’s Mean Value Theorem
A generalized mean-value relation involving two functions. \[\frac{f'(c)}{g'(c)}=\frac{f(b)-f(a)}{g(b)-g(a)}\]
6.15 Fermat’s theorem
An interior differentiable local extremum has zero derivative. \[\text{Local extremum at c } \Longrightarrow f'(c)=0\]
6.16 Darboux theorem
Derivatives have the intermediate value property even though they need not be continuous. \[f' \text{ has IVP}\]
6.17 Derivative bounded \(\Longrightarrow\) Lipschitz
If |f’|\(\le\)M on an interval, f is Lipschitz there. \[|f'(x)|\le{}M \Longrightarrow |f(x)-f(y)|\le{}M|x-y|\]
Applications of Single-Variable Derivatives
7.1 Critical point
A critical point is an interior point where f’=0 or f’ does not exist. \[f'(c)=0 \text{ or } f'(c) \text{ DNE }\]
7.2 Local maximum
f(c) is at least as large as nearby function values. \[f(c)\ge{}f(x) \text{ near } c\]
7.3 Local minimum
f(c) is at most as large as nearby function values. \[f(c)\le{}f(x) \text{ near } c\]
7.4 First derivative test
A change from positive to negative gives a local maximum; negative to positive gives a local minimum. \[+\to- : max; \:\:\: -\to+ : min\]
7.5 Monotonicity theorem
If f’\(\ge\)0 throughout an interval, f is nondecreasing; if f’\(\le\)0, f is nonincreasing. \[f'\ge0 \Longrightarrow f \text{ non-decreasing.}\]
7.6 Strict monotonicity theorem
If f’>0 throughout an interval, f is strictly increasing; similarly for f’<0. \[f'>0 \Longrightarrow f \text{ strictly increasing.}\]
7.7 Second derivative test
At a stationary point, positive second derivative gives a local minimum and negative gives a local maximum. \[f'(c)=0; \:\:\: f''(c)>0 \Longrightarrow min; \:\:\: f''(c)<0 \Longrightarrow max\]
7.8 Concavity
Positive second derivative corresponds to concavity upward; negative to downward. \[f''>0 \Longrightarrow \text{ convex/upward;} \:\: f''<0 \Longrightarrow \text{ concave/downward.}\]
7.9 Inflection point
A point where the concavity changes is an inflection point, subject to the standard local definition. \[\text{ Concavity changes at c }\Longrightarrow \text{ Inflection point.}\]
7.10 Generalized second derivative test
If the first nonzero derivative after f’ at c is of even order, its sign determines max/min; odd order gives no extremum. \[f'(c)=\cdots=f^{(m-1)}(c)=0, f^{(m)}(c)\ne0\]
7.11 Taylor theorem
A smooth function can be approximated by a polynomial with a controlled remainder. \[f(x) = \sum_{k=0}^{n} \frac{f^{(k)}(a)}{k!} (x-a)^k + R_{n}(x)\]
7.12 Lagrange remainder
Taylor’s remainder has a derivative-based representation. \[R_{n}(x) = \frac{f^{(n+1)}(\xi)}{(n+1)!} (x-a)^{n+1}\]
7.13 Maclaurin expansion
Taylor expansion about zero. \[f(x) = \sum_{k=0}^{\infty} \frac{f^{(k)}(0)}{k!} x^k\]
7.14 L’Hopital’s rule
Under standard hypotheses, 0/0 or \(\infty\)/\(\infty\) limits may be evaluated using derivative ratios. \[lim \frac{f}{g}=lim\frac{f'}{g'}\]
7.15 Cauchy MVT consequence
If f’=0 on an interval, f is constant there. \[f'=0 \Longrightarrow f is constant\]
7.16 Mean-value bound
A bounded derivative controls function differences. \[|f'|\le{}M \Longrightarrow |f(b)-f(a)|\le{}M|b-a|\]
7.17 Inverse-function monotonicity
A continuous strictly monotone function has an inverse on its range, and the inverse is continuous. \[f \text{ strictly monotone } \Longrightarrow f^{-1} \text{ exists on } f(I).\]
7.18 Optimization principle
A continuous function on a closed bounded interval attains its absolute extrema; candidates include endpoints and critical points. \[\text{The set of candidate points for local or absolute extrema on } [a, b] \text{ is given by:}\] \[E = \{a, b\} \cup \{c \in (a, b) : f'(c) = 0\} \cup \{c \in (a, b) : f'(c) \text{ does not exist}\}\]
7.19 Rolle-based root uniqueness
If f’ never vanishes on an interval, f has at most one zero there. \[f'\ne0 \Longrightarrow \text{ At most one root.}\]
Riemann Integration
8.1 Partition
A partition of [a,b] divides it into finitely many subintervals. \[P={a=x_{0}<x_{1}<\cdots<x_{n}=b}\]
8.2 Mesh/norm of partition
The mesh is the length of the largest subinterval. \[||P||=max_i(x_i-x_{i-1})\]
8.3 Upper sum
The upper Darboux sum uses the supremum of f on each subinterval. \[U(f,P)=\sum M_i\Delta x_i\]
8.4 Lower sum
The lower Darboux sum uses the infimum on each subinterval. \[L(f,P)=\sum m_i\Delta x_i\]
8.5 Upper integral
The upper integral is the infimum of upper sums. \[\overline{\int_{a}^{b}} f(x) \, dx = \inf_{P} U(f, P)\]
8.6 Lower integral
The lower integral is the supremum of lower sums. \[\underline{\int_{a}^{b}} f(x) \, dx = \sup_{P} L(f, P)\]
8.7 Riemann integrability criterion
A bounded function is Riemann integrable iff its upper and lower integrals agree. \[f \text{ is Riemann integrable on } [a, b] \iff \underline{\int_{a}^{b}} f(x) \, dx = \overline{\int_{a}^{b}} f(x) \, dx\]
8.8 Darboux criterion
A bounded function is integrable iff upper and lower sums can be made arbitrarily close. \[\forall\varepsilon>0, \:\: \exists{} \:\: P: U(f,P)-L(f,P)<\varepsilon\]
8.9 Continuous functions are integrable
Every continuous function on [a,b] is Riemann integrable. \[f\in{}C[a,b] \Longrightarrow f\in{}R[a,b]\]
8.10 Monotone functions are integrable
Every bounded monotone function on a closed interval is Riemann integrable. \[f \text{ monotone on }[a,b] \Longrightarrow f\in{}R[a,b]\]
8.11 Basic integral properties
Riemann integrals are linear and additive over intervals. \[\int(\alpha{}f+\beta{}g)=\alpha\int f+\beta\int{}g\]
8.12 Order property
If f\(\le\)g, then their integrals satisfy the same order. \[f\le{}g \Longrightarrow \int f\le\int{}g\]
8.13 Absolute-value inequality
The integral of the absolute value dominates the absolute value of the integral. \[\left| \int f \right|\le\int|f|\]
8.14 Bound estimate
A bounded function has an integral bounded by its supremum times interval length. \[|f|\le{}M \Longrightarrow \left|\int_a^bf\right|\le{}M(b-a)\]
8.15 Additivity over intervals
Integrals can be split at an intermediate point. \[\int_a^bf=\int_a^cf+\int_c^b f\]
8.16 Fundamental Theorem of Calculus I
If f is continuous, its accumulation function has derivative f. \[F(x)=\int_a^xf(t)dt \Longrightarrow F'(x)=f(x)\]
8.17 Fundamental Theorem of Calculus II
A differentiable antiderivative evaluates a definite integral by endpoint values. \[\int_a^bf(x)dx=F(b)-F(a), F'=f\]
8.18 Integration by parts
The integral of a product can be transformed using derivatives of the factors. \[\int{}u dv=uv-\int{}v du\]
8.19 Substitution theorem
A differentiable change of variable transforms the integral using the derivative of the substitution. \[\int f(g(x))g'(x)dx=\int f(u)du\]
8.20 Mean Value Theorem for integrals
A continuous function assumes its average value somewhere in the interval. \[\exists{}\:c\in[a,b]: \int_a^bf=f(c)(b-a)\]
8.21 Integral average value
The average value of f over [a,b] is the integral divided by interval length. \[f_{\text{avg}} = \frac{1}{b - a} \int_{a}^{b} f(x) \, dx\]
8.22 Improper integral
An integral with an infinite interval or unbounded integrand is defined through a limit. \[\int_a^\infty f=\lim_{b\to^\infty}\int_a^bf\]
8.23 p-integral at infinity
The integral \(\int_{1}^{\infty}x^{-p}\,\,\mathrm{d}x\) converges exactly for p>1. \[\int_{1}^{\infty} \frac{1}{x^{p}} \, dx \text{ converges} \iff p > 1\]
8.24 Comparison test for improper integrals
Nonnegative functions can be compared to establish convergence or divergence. \[0\le{}f\le{}g, \:\:\:\int{}g<\infty \Longrightarrow \int f<\infty\]
Standard Limits, Series and Formulas
9.1 Exponential series
Maclaurin expansion of \(e^x\). \[e^{x} = \sum_{n=0}^{\infty} \frac{x^{n}}{n!}\]
9.2 Sine series
Maclaurin expansion of sin x. \[\sin x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n+1}}{(2n+1)!}\]
9.3 Cosine series
Maclaurin expansion of cos x. \[\cos x = \sum_{n=0}^{\infty} \frac{(-1)^{n} x^{2n}}{(2n)!}\]
9.4 Logarithm series
For |x|<1. \[\ln(1+x) = \sum_{n=1}^{\infty} \frac{(-1)^{n+1} x^{n}}{n}\]
9.5 Binomial series
For suitable \(\alpha\) \[(1+x)^{\alpha} = \sum_{n=0}^{\infty} \binom{\alpha}{n} x^{n}, \quad \text{for } |x| < 1\]
9.6 Standard sine limit
A fundamental limit used in trigonometric differentiation. \[\lim_{x \to 0} \frac{\sin x}{x} = 1\]
9.7 Standard cosine limit
A fundamental second-order trigonometric limit. \[\lim_{x \to 0} \frac{1 - \cos x}{x^{2}} = \frac{1}{2}\]
9.8 Exponential limit
A standard definition-compatible exponential limit. \[\lim_{x \to 0} \frac{e^{x} - 1}{x} = 1\]
9.9 Logarithmic limit
A standard logarithmic limit. \[\lim_{x \to 0} \frac{\ln(1 + x)}{x} = 1\]
9.10 Euler exponential limit
A classical limit defining e. \[\lim_{n \to \infty} \left(1 + \frac{1}{n}\right)^{n} = e\]
9.11 Power-exponential limit
For suitable real a. \[\lim_{x \to \infty} \left(1 + \frac{a}{x}\right)^{x} = e^{a}\]
Important Inequalities and Consequences
10.1 AM-GM inequality
For nonnegative a,b, arithmetic mean dominates geometric mean. \[\frac{a + b}{2} \ge \sqrt{ab}\]
10.2 Cauchy-Schwarz inequality
The square of an inner product is at most the product of squared norms. \[\left( \sum_{i} a_i b_i \right)^{2} \le \left( \sum_{i} a_i^2 \right) \left( \sum_{i} b_i^2 \right)\]
10.3 Young’s inequality
For conjugate exponents p,q>1. \[a b \le \frac{a^{p}}{p} + \frac{b^{q}}{q}, \quad \text{where } \frac{1}{p} + \frac{1}{q} = 1 \text{ and } p, q > 1\]
10.4 Bernoulli inequality
For n\(\in\)\(\mathbb{N}\) and x\(\ge\)\(-\)1. \[(1+x)^{n}\ge1+nx\]
10.5 Jensen’s inequality
A convex function of a weighted average does not exceed the weighted average of function values. \[f\left( \sum_{i=1}^{n} \lambda_i x_i \right) \le \sum_{i=1}^{n} \lambda_i f(x_i)\]
10.6 Convexity criterion
A twice differentiable function is convex when its second derivative is non-negative. \[f''\ge0 \Longrightarrow f \text{ convex.}\]
10.7 Concavity criterion
A twice differentiable function is concave when its second derivative is non-positive. \[f''\le0 \Longrightarrow f \text{ concave.}\]
10.8 Young/Holder family
The classical inequalities relate products and sums under conjugate exponents and lead to Holder-type bounds.
10.9 Integral Cauchy-Schwarz
For square-integrable functions, the integral inner product satisfies Cauchy-Schwarz. \[\left|\int fg\right|\le\left(\int f^2\right)^{1/2}\left(\int g^2\right)^{1/2}\] In detail, \[\left| \int_{a}^{b} f(x)g(x) \, dx \right| \le \left( \int_{a}^{b} f(x)^2 \, dx \right)^{1/2} \left( \int_{a}^{b} g(x)^2 \, dx \right)^{1/2}\]