Power Series Calculator

Ask and get solution to your homeworkAsk now and get step-by-step solutions

Example

Created on 2024-06-20Asked by Mila Carter (Solvelet student)

Find the power series representation for the function

f(x) = \frac{1}{1 - x}

Solution

To find the power series representation for the function

f(x) = \frac{1}{1 - x}

: 1. Recognize that

\frac{1}{1 - x}

is a geometric series with first term

a = 1

and common ratio

r = x

. 2. Recall the formula for the sum of an infinite geometric series:

S = \frac{a}{1 - r},

where

|r| < 1

. 3. Substitute the values of

a

and

r

into the formula:

f(x) = \frac{1}{1 - x} = \frac{1}{1 - (-x)} = \sum_{n=0}^{\infty} (-x)^n.

Therefore, the power series representation for

f(x)

\sum_{n=0}^{\infty} (-x)^n

. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Theodore Rivera on Solvelet

1. Find the interval of convergence of the power series

\sum_{n=0}^\infty \frac{(x - 1)^n}{n^2}

.2. Determine whether the power series

\sum_{n=0}^{\infty} \frac{(x - 2)^n}{2^n}

converges or diverges.

DefinitionA power series is an infinite series of the form+∞∑n=0a n (x−c)nwhere an, are coefficients, and c is the centre of the series. Power series converge to a function within a disk of convergence.

Related Calculators

Homeomorphisms Calculator Express in Terms of Sine and Cosine Calculator Curve Sketching Calculator Numerical Methods Calculator

Need topic explanation ? Get video explanation