Representing Functions as Power Series

Introduction to representing functions as power series:

A power series in one variable is an infinite sequence of the structure,

f(x) = `sum_(n=0)^oo a_(n)(x-c)^(m+n) = a_(0)+a_(1)(x-c)^(1)+a_(2)(x-c)^2+...`  where an correspond to the coefficient of the nth expression, c is a constant, and x varies about c. This series usually occur as the Taylor series of some recognized function the Taylor series.

In several situations c is equal to zero, for instance when allowing for a Maclaurin series. In such cases, the power series obtain the simpler structure   f(x) = `sum_(n=0)^oo a_(n)(x)^(n) = a_(0)+a_(1)(x)+a_(2)(x)^2+...`

Representing Functions as Power Series:

Power series arise in combinatory in the name of produce functions in the name of the Z-transform. The identifiable decimal details for actual numbers recognize how to also be analysis as an example of a power series, with integer coefficients, but with the case x set at 1⁄10.

Power series are calculation a generality of polynomials as formal substance, wherever the number of expressions is allowed to exist unlimited. This involve give up the option to reserve subjective values for indefinite.

This analysis contrast with of power series, whose variables assign arithmetical values, and to series so only include a specific value if junction knows how to be recognized. Is this topic how many faces does a cylinder have hard for you? Watch out for my coming posts.

Examples for Representing Functions as Power Series:

Example 1:

How to solve representing function as power series `1/(1-x^2)`

Solution:

Step 1: the given function is `1/(1-x^2)`

Step 2: to evaluate the function is

`sum_(n=0)^oo(x^2)^n`

Step 3:   `|x^2| <1 br="br">
Step 4:   `|x|^2 <1 br="br">
Step 5: so the solution is `-1
Example 2:

How to solve representing function as power series `1/(1-9x^2)`

Solution:

Step 1: the given function is `1/(1-9x^2)`

Step 2: to evaluate the function is

`sum_(n=0)^oo(9x^2)^n`

Step 3:   `|9x^2| <1 br="br">
Step 4:   `|x|^2 <1 br="br">
Step 5: so the solution is `-1/3
Example 3:

How to solve representing function as power series `x/(4x-1)`

Solution:

Step 1: the given function is   `x/(4x-1)`

`x(1/(1-4x))`

Step 2: to evaluate the function is

`xsum_(n=0)^oo(4x)^n`

Step 3:     `xsum_(n=0)^oo(4)^n(x)^n`

Step 4:      `sum_(n=0)^oo(4)^n(x)^n-x`

Step 5:           `sum_(n=0)^oo(4)^n(x)^(n+1)`

So the solution is      `sum_(n=0)^oo(4)^n(x)^(n+1)`

Example 4:

How to solve representing function as power series `1/(1-16x^2)`

Solution:

Step 1: the given function is `1/(1-16x^2)`

Step 2: to evaluate the function is

`sum_(n=0)^oo(16x^2)^n`

Step 3:   `|16x^2| <1 br="br">
Step 4:   `|x|^2 <1 br="br">
Step 5: so the solution is `-1/4