Infinite Series

Tags: 
sequence
series
finite series
infinites series
standard series
general term
logarithmic series
binomial series
exponential series
hyperbolic series
trigonometric 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!} + ... $
 

Submitted by Romel Verterra on March 12, 2012 - 8:28pm
  • 11413 reads
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