SEQUENCES AND SERIES

Finite Series: expression of the form [pic]where each term is formed by some definite rule.
Infinite Series: it is of the form [pic] (Example: infinite geometric series with a0=1 and r=1/2) [pic]

Sum of Infinite Series (Sn) ❖ The sum of finite series is merely the algebraic sum of all term. (addition) ❖ Infinite series has no sum in ordinary sense.

Definitions: The sum of an infinite series is the limit, if it exist, of the sum of a finite number of terms, as the number of terms approaches infinity. ([pic]
(Sum of infinite geometric progression)

Convergence and Divergence
If the series has a sum Sn i.e. if Sn approaches a limit when [pic]

Power Series

An infinite series of the form [pic] (1)

in which a0 , a1, a2, . . . , an, . . . are constants and x is a variable, is called a power series in x. The totality of all values of x for which a power series converges is called its interval of convergence. This interval of convergence always includes the value x = 0, and its range is determined by the ratio test, R = [pic].
Examples. Find the interval of convergence for each of the following series.
1. [pic]
2. [pic]
3. [pic]
4. [pic]
5. [pic]
6. [pic]

Maclaurin’s Series

A power series of the form [pic]. is called Maclaurin’s series, or the power-series expansion of f(x) about x = 0. It is named for the Scottish mathematician Colin Maclaurin (1698 – 1746).

Examples. Obtain the first four terms of the Maclaurin’s series for the following functions. 1. f(x) = ex 2. f(x) = sin x 3. f(x) = cos x 4. f(x) = ln(x+1) 5. f(x) = 1/(1+x)
Some power series expansions: [pic] for all x [pic] for all x [pic] for [pic]

[pic]for -1 < x < 1 [pic] for [pic] [pic] for all x [pic] for all x

Taylor’s Series

A power series expansion of f(x) about x = a is of the form

[pic].

This series is called Taylor’s series, named for the English mathematician Brook Taylor (1685 – 1731).

Examples. Obtain the first four terms of the Taylor’s series expansion for the given functions about the indicated point. 1. f(x) = cos x about x = (/4 2. f(x) = ex about x = -1 3. f(x) = ln x about x = 2 4. [pic] about x = 1 5. f(x) = tan x about x = (/4

Find the Taylor series expansion of
1. [pic] about the point x = 4.

Ans. [pic]
2. f(x) = tan x about x = [pic]

3. f(x) = [pic] about x = 1

4. f(x)=[pic] in powers of x - 1