• TODO
• Series convergence
  • Summary
  • Root test
  • Telescoping test
  • Cauchy condensation test
  • Abel’s test
• Misc
  • Bernoulli’s inequality
  • More
• Sequences
• Limits
• Gradient
• Questions

TODO

The Taylor series of $f(x)$ that is infinitely differentiable in a neighborhood of $a$
$\begin{align*} &f(a)+\frac {f^{(1)}(a)}{1!} (x-a)+ \frac{f^{(2)}(a)}{2!} (x-a)^2+\frac{f^{(3)}(a)}{3!}(x-a)^3+ \cdots = \\ \\ &=\sum_{n=0}^{\infty} \frac {f^{(n)}(a)}{n!} \, (x-a)^{n} \end{align*}$

Maclaurin series of $e^x$
$e^{x} = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \cdots + \frac{x^n}{n!} + o(x^n)$

Maclaurin series of $\sin x$
$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + o(x^8)$

$$\sin x = x - \frac{x^3}{3!} + \frac{x^5}{5!} - \frac{x^7}{7!} + \frac{x^9}{9!} + o(x^{10})$$

Maclaurin series of $\cos x$
$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} + o(x^9)$

$$\cos x = 1 - \frac{x^2}{2!} + \frac{x^4}{4!} - \frac{x^6}{6!} + \frac{x^8}{8!} - \frac{x^{10}}{10!} + o(x^{11})$$

Maclaurin series of $\ln (1+x)$
$\ln(1+x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + o(x^4)$

Maclaurin series of $(1+x)^m$
$\begin{align*} (1+x)^m &= 1 + mx + \frac{m(m-1)}{2!} x^2 + \frac{m(m-1)(m-2)}{3!} x^3 + \cdots + \\ \\ &+ \frac{m(m-1)\cdots(m-n+1)}{n!} x^n + o(x^n) \end{align*}$

$(e^x)’ = \ ?$
$(e^x)’ = e^x$

$(1)’ = \ ?$
$(1)’ = 0$

$(x^a)’ = \ ?$
$(x^a)’ = a\; x^{a-1}$

$(\log_a x)’ = \ ?$
$(\log_a x)’ = \frac{1}{x \ln a}$

$(\ln x)’ = \ ?$
$(\ln x)’ = \frac{1}{x}$

$(\sin x)’ = \ ?$
$(\sin x)’ = \cos x$

$(\cos x)’ = \ ?$
$(\cos x)’ = - \sin x$

$(\operatorname{tg} x)’ = \ ?$
$(\operatorname{tg} x)' = \frac{1}{\cos^2 x}$

$(\arcsin x)’ = \ ?$
$(\arcsin x)' = \frac{1}{\sqrt{1-x^2}}$

$(\arccos x)’ = \ ?$
$(\arccos x)' = \frac{-1}{\sqrt{1-x^2}}$

$(\operatorname{arctg} x)’ = \ ?$
$(\operatorname{arctg} x)’ = \frac{1}{1+x^2}$

$(af + bg)’ = \ ?$
$(af + bg)’ = af’ + bg’$

$(fg)’ = \ ?$
$(fg)’ = f ‘g + fg’$

$\left(\frac{f}{g} \right)’ = \ ?$
$\left(\frac{f}{g} \right)’ = \frac{f’g - fg’}{g^2}$

$\left(f (g) \right)’ = \ ?$
$\left(f (g) \right)’ = f’(g) \cdot g’$

$\int x^{\alpha} \, dx = \ ?$
$\int x^{\alpha} \, dx \ = \ \frac{x^{\alpha+1}}{\alpha+1}, \ \ \ \ \ \alpha \neq -1$

$$\int x^{-1} \, dx \ = \int \frac{1}{x} \, dx \ = \ \ln \vert x \vert$$

$\int \frac{1}{x} \, dx = \ ?$
$\int \frac{1}{x} \, dx \ = \ \ln \vert x \vert$

$\int a^x \, dx = \ ?$
$\int a^x \, dx \ = \ \frac{a^x}{\ln a}$

$\int e^x \, dx = \ ?$
$\int e^x \, dx \ = \ e^x$

$\int \sin x \; dx = \ ?$
$\int \sin x \; dx \ = \ - \cos x$

$$\int \cos x \; dx = \ ?$$ $$\int \cos x \; dx \ = \ \sin x$$

$\int \frac{1}{\cos^2 x} \; dx = \ ?$
$\int \frac{1}{\cos^2 x} \; dx \ = \ \operatorname{tg} x$

$\int \frac{1}{\sin^2 x} \; dx = \ ?$
$\int \frac{1}{\sin^2 x} \; dx \ = \ - \operatorname{ctg} x$

$\int \frac{1}{a^2 + x^2} \; dx = \ ?$
$\int \frac{1}{a^2 + x^2} \; dx \ = \ \frac{1}{a} \; \operatorname{arctg} \frac{x}{a}$

$\int \frac{1}{a^2 - x^2} \; dx = \ ?$
$\int \frac{1}{a^2 - x^2} \; dx \ = \ \frac{1}{2a} \; \ln \left \vert \frac{a+x}{a-x} \right \vert$

$\int \frac{1}{ \sqrt{a^2-x^2} } \; dx = \ ?$
$\int \frac{1}{ \sqrt{a^2-x^2} } \; dx \ = \ \arcsin \frac{x}{a}$

$\int \frac{1}{\sqrt{x^2 + a}} \; dx = \ ?$
$\int \frac{1}{\sqrt{x^2 + a}} \; dx \ = \ \ln \left \vert x+\sqrt{x^2 + a} \right \vert$

integration by parts

$\int u\, dv = \ ?$
$\int u \; dv \ = \ uv \ - \ \int v \; du$

integration by parts

$\int\limits_a^b u \; dv = \ ?$
$\int\limits_a^b u \; dv \ = \ u\,v \ \bigg \vert _a^b \ - \ \int\limits_a^b v \; du$

integration by substitution

$$\int f(x) \; dx = F(x)$$

$\int f\left( \varphi(x) \right) \; d \varphi(x) = \ ?$
$\int f\left( \varphi(x) \right) \; d \varphi(x) \ = \ F\left( \varphi(x) \right)$

$\int a \; f(x) \; dx = \ ?$
$\int a \; f(x) \; dx \ = \ a \int f(x) \; dx$

$\int \left[ f(x) \pm g(x) \right] \; dx = \ ?$
$\int \left[ f(x) \pm g(x) \right] \; dx \ = \ \int f(x) \; dx \ \pm \ \int g(x) \; dx$

$\int 0\; dx = \ ?$
$\int 0\; dx \ = \ 0 \ + \ C$

$\int a \; dx = \ ?$
$\int a \; dx \ = \ ax$

$\int e^{-x^2}\; dx = \ ?$
$\int e^{-x^2}\; dx \ = \ \text{special function} \ (x)$

$\int e^{x^2}\; dx = \ ?$
$\int e^{x^2}\; dx \ = \ \text{special function} \ (x)$

$\int \frac{1}{\ln x} \; dx = \ ?$
$\int \frac{1}{\ln x} \; dx \ = \ \text{special function} \ (x)$

$\int \sin x^2 \; dx = \ ?$
$\int \sin x^2 \; dx \ = \ \text{special function} \ (x)$

$\int \cos x^2 \; dx = \ ?$
$\int \cos x^2 \; dx \ = \ \text{special function} \ (x)$

$\int \frac{\sin x}{x} \; dx = \ ?$
$\int \frac{\sin x}{x} \; dx \ = \ \text{special function} \ (x)$

$\int \frac{\cos x}{x} \; dx = \ ?$
$\int \frac{\cos x}{x} \; dx \ = \ \text{special function} \ (x)$

$\int \frac{e^x}{x} \; dx = \ ?$
$\int \frac{e^x}{x} \; dx \ = \ \text{special function} \ (x)$

$\int \operatorname{sh} x \; dx = \ ?$
$\int \operatorname{sh} x \; dx \ = \ \operatorname{ch} x$

$\int \operatorname{ch} x \; dx = \ ?$
$\int \operatorname{ch} x \; dx \ = \ \operatorname{sh} x$

$\int \frac{1}{\operatorname{sh}^2 x} \; dx = \ ?$
$\int \frac{1}{\operatorname{sh}^2 x} \; dx \ = \ - \mathrm{cth} x$

$\int \frac{1}{\operatorname{ch}^2 x} \; dx = \ ?$
$\int \frac{1}{\operatorname{ch}^2 x} \; dx \ = \ \mathrm{th} x$

five special functions in calculus

$$\int \frac{1}{\ln x} \; dx$$ $$\int e^{x^2}\; dx$$ $$\int \sin x^2 \; dx$$ $$\int \frac{e^x}{x} \; dx$$ $$\int \frac{\sin x}{x} \; dx$$

$\int \frac{1}{\sqrt{x^2 + a^2}} \; dx = \ ?$
$\int \frac{1}{\sqrt{x^2 + a^2}} \; dx \ = \ \ln \left \vert x+\sqrt{x^2 + a^2} \right \vert$

$\int \frac{1}{\sqrt{x^2 - a^2}} \; dx = \ ?$
$\int \frac{1}{\sqrt{x^2 - a^2}} \; dx \ = \ \ln \left \vert x+\sqrt{x^2 - a^2} \right \vert$

Series convergence

https://en.wikipedia.org/wiki/Convergence_tests

http://www.math.hawaii.edu/~ralph/Classes/242/SeriesConvTests.pdf
http://www.toomey.org/tutor/harolds_cheat_sheets/Harolds_Series_Convergence_Tests_Cheat_Sheet_2016.pdf

http://tutorial.math.lamar.edu/Classes/CalcII/SeriesStrategy.aspx
http://www.furius.ca/cqfpub/doc/series/series.pdf
http://www.toomey.org/tutor/harolds_cheat_sheets/Harolds_Calculus_Notes_Cheat_Sheet_2017.pdf
https://www.math.wvu.edu/~hjlai/Teaching/Math156_Website/Series Cheat Sheet.pdf
https://www.math.hmc.edu/calculus/tutorials/convergence/

http://infotables.ru/matematika/66-ryady/628-priznaki-skhodimosti-chislovogo-ryada-tablitsa

Summary

Nesessary condition: if $\lim_{n \to \infty}a_n \ne 0$ then diverges

Absolute convergence: does $\sum_{n=0}^\infty \left \vert a_n\right \vert$ converge?

Conditional convergence

Ratio test: $\lim _{n\to \infty }\left \vert {\frac {a_{n+1}}{a_{n}}}\right \vert =r$

Root test: $\lim_{n\rightarrow\infty}\sqrt[n]{ \vert a_n \vert }$

Direct comparison: $ \vert a_n \vert \le \vert b_n \vert $

Limit comparision: $\lim _{n\to \infty }{\frac {a_{n}}{b_{n}}}$

Integral test: $\int _{1}^{\infty }f(x)\,dx=\lim _{t\to \infty }\int _{1}^{t}f(x)\,dx<\infty$

Abel’s test: $\sum a_nb_n$

Cauchy condensation test: $\sum_{n=0}^\infty 2^n a_{2^n}$

Harmonic series: $\sum n^p$ converges if $p>1$, diverges if $p \le 1$

Geometric series: $\sum \left( 1 + \frac{1}{q} + \frac{1}{q^2} + \frac{1}{q^3} \ldots \right) = \frac{1}{1-q}$, when $\vert q \vert \ < 1$

Alternating series $\sum _{n=k}^{\infty }(-1)^{n}a_{n}$

Telescoping test: $\sum_{n=1}^\infty \frac{1}{n(n+1)} = \sum_{n=1}^\infty \left( \frac{1}{n} - \frac{1}{n+1} \right)$, summands cancel out in partial sums.

Taylor series test: is it Taylor series?

Root test

https://en.wikipedia.org/wiki/Root_test

$\lim_{n\rightarrow\infty}\sqrt[n]{ \vert a_n \vert }$ or $r=\limsup_{n\to \infty }{\sqrt[{n}]{ \vert a_{n} \vert }}$

Telescoping test

https://en.wikipedia.org/wiki/Telescoping_series

https://math.stackexchange.com/questions/104918/how-to-analyze-convergence-and-sum-of-a-telescopic-series-i-cant-find-a-generi

Cauchy condensation test

https://en.wikipedia.org/wiki/Cauchy_condensation_test

Abel’s test

https://en.wikipedia.org/wiki/Abel's_test

Misc

Bernoulli’s inequality

It approximates exponentiation of $1+x$:

$(1+x)^n \geq 1+nx$ for $x \geq -1$

Or this:

$$(1+x_1)(1+x_2)\cdots(1+x_n) \ge 1+x_1+x_2+\ldots+x_n$$

More

Sequences

$$\sum \left( 1 + 2 + 3 + \ldots + n \right) = \frac{(n+1)n}{2}$$ $$\sum \left( 1 + 2 + 2^2 + 2^3 + 2^4 + 2^5 + \ldots + 2^{n-1} \right) = 2^n - 1$$ $$\sum \left( 1 + 2^2 + 3^2 + 4^2 + \ldots + n^2 \right) = \frac{n(n+1)(2n+1)}{6}$$ $$\sum \left( 1 + 2^3 + 3^3 + 4^3 + \ldots + n^3 \right) = \left( \frac{(n+1)n}{2} \right)^2$$

$\sum \left( 1 + 2^k + 3^k + 4^k + \ldots + n^k \right)$ is a polynomial $p(k)$ of degree $k+1$

Limits

Harmonic series, https://en.wikipedia.org/wiki/Harmonic_series_(mathematics)#Divergence:

$$1 + \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \frac{1}{5} + \ldots = \infty$$

Alternating harmonic series — special case of Taylor series of logarithm; it’s conditionally convergent, not absolutely though, if rearranged the sum becomes different:

$$1 - \frac{1}{2} + \frac{1}{3} - \frac{1}{4} + \frac{1}{5} + \ldots = \ln 2$$

https://en.wikipedia.org/wiki/Telescoping_series

Gradient

Magnitude of gradient is equal to tan of angle of tangent plane.

Questions