Infinite Series

Sequences and Series

Sequence is a succession of numbers formed according to some fixed rule. Example is

$1,~ 8,~ 27,~ 64,~ 125,~ ...$

which is a sequence so that the nth term is given by n3.
 

Series is the indicated sum of a sequence of numbers. Thus,

$a_1 + a_2 + a_3 + ... + a_n + ...$

is the series corresponding to the sequence $a_1,~ a_2,~ a_3,~ ... ,~a_n,~ ...$
 

Finite and Infinite Series
A series is said to be finite if the number of terms is limited. It is infinite series if the number of terms is unlimited.

General Term of a Series
The general term of a series is an expression involving n, such that by taking n = 1, 2, 3, ..., one obtains the first, second, third, ... term of the series.
 

Standard Series

Binomial Series
1. $(a + b)^n = a^n + na^{n-1}b + \dfrac{n(n-1)}{2!}a^{n-2}b^2 + \dfrac{n(n-1)(n-2)}{3!}a^{n-3}b^3 + ...$

2. $\dfrac{1}{a - bx} = \dfrac{1}{a} \left( 1 + \dfrac{bx}{a} + \dfrac{b^2x^2}{a^2} + \dfrac{b^3x^3}{a^3} + ... \right)$
 

Trigonometric Series
1. $\sin x = x - \dfrac{x^3}{3!} + \dfrac{x^5}{5!} - \dfrac{x^7}{7!} + ...$

2. $\cos x = 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{4!} - \dfrac{x^6}{6!} + ...$

3. $\tan x = x = \dfrac{x^3}{15} + \dfrac{x^5}{15} + \dfrac{17x^7}{315} + ...$

4. $\sec x = 1 + \dfrac{x^2}{2} + \dfrac{5x^4}{24} + \dfrac{61x^6}{720} + ...$

5. $\arcsin x = x + \dfrac{x^3}{6} + \dfrac{3x^5}{40} + ...$

6. $\arctan x = x - \dfrac{x^3}{3} + \dfrac{x^5}{5} - \dfrac{x^7}{7} + ...$

7. $\sin 2x = 2x - \dfrac{4x^3}{3} + \dfrac{4x^5}{15} - \dfrac{8x^7}{315} + ...$

8. $\cos^2 x = 1 - x^2 + \dfrac{x^4}{3} - \dfrac{2x^6}{45} + ...$

9. $\cos x^2 = 1 - \dfrac{x^4}{2!} + \dfrac{x^8}{4!} - \dfrac{x^12}{6!} - ...$

10. $\cos 2x = 1 - 2x^2 + \dfrac{2x^4}{3} + ...$
 

Exponential Series
1. $e^x = 1 + x + \dfrac{x^2}{2!} + \dfrac{x^3}{3!} + \dfrac{x^4}{4!} + ...$

2. $e^{-x^2} = -x^2 + \dfrac{x^4}{2!} - \dfrac{x^6}{3!} + \dfrac{x^8}{4!} - ...$

3. $e^{\sin x} = 1 + x + \dfrac{x^2}{2!} - \dfrac{3x^4}{4!} - \dfrac{8x^5}{5!} + \dfrac{3x^6}{6!} + ...$

4. $e^{\cos x} = e\left( 1 - \dfrac{x^2}{2!} + \dfrac{x^4}{6!} - \dfrac{31x^6}{720} + ... \right)$

5. $e^{\tan x} = 1 + x + \dfrac{x^2}{2!} + \dfrac{3x^3}{3!} + \dfrac{9x^4}{4!} + \dfrac{37x^5}{5!} + ...$
 

Logarithmic Series
1. $\ln x = (x - 1) - \dfrac{(x - 1)^2}{2} + \dfrac{(x - 1)^3}{3} - ...$

2. $\ln (x + 1) = x - \dfrac{x^2}{2} + \dfrac{x^3}{3} - \dfrac{x^4}{4} + ...$

3. $\ln \left( \dfrac{1 + x}{1 - x} \right) = 2\left(x + \dfrac{x^3}{3} + \dfrac{x^5}{5} + \dfrac{x^7}{7} + ... \right)$

4. $\ln \left( \dfrac{x + 1}{x - 1} \right) = 2 \left( \dfrac{1}{x} + \dfrac{1}{3x^3} + \dfrac{1}{5x^5} + \dfrac{1}{7x^7} + ... \right)$

5. $\ln \sin x = \ln x - \dfrac{x^2}{6} - \dfrac{x^4}{180} - \dfrac{x^6}{2835} - ...$

6. $\ln \cos x = -\dfrac{x^2}{2} - \dfrac{x^4}{12} - \dfrac{x^6}{45} - \dfrac{17x^8}{2520} - ...$

7. $\ln \tan x = \ln x + \dfrac{x^2}{3} + \dfrac{7x^4}{90} + \dfrac{62x^6}{2835} + ...$

8. $\ln (1 + \sin x) = x - \dfrac{x^2}{2} + \dfrac{x^3}{6} - \dfrac{x^4}{12} + ...$
 

Hyperbolic Series
1. $\sinh x = \frac{1}{2}(e^x - e^{-x}) = x + \dfrac{x^3}{3!} + \dfrac{x^5}{5!} + ...$

2. $\cosh x = \frac{1}{2}(e^x + e^{-x}) = 1 + \dfrac{x^2}{2!} + \dfrac{x^4}{4!} + ...$
 

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