Q-25E - Fundamentals Of Differential Equations And Boundary Value Problems

Question

Question: In Problems 23–26, express the given power series as a series with generic term X^K.

25.$\sum _{n=0}^{\infty }{a}_n{x}^{n+1}$ 

Step-by-Step Solution

Answer

The required term is $\sum _{k=1}^{\infty }{a}_{k-1}{x}^k.$

Step 1: Power series

A power series is an infinite series of the form,

$\sum _{n=0}^{\infty }{a}_n{(x-c)}^n=a_0+ a_1(x-c) +a_2(x-c{)}^2+.....$

Where,a_n represents the coefficient term of the nth term and c is a constant.

Step 2: To express the given series in terms of a generic term x k

In order to express the given series in terms of generic term x^k , we will change the index of the power series.

Given that, $f\left(x\right)=\sum _{n=0}^{\infty }{a}_n{x}^{n+1}$

Let,

$$\begin{gathered} n+1=k \\ n=k - 1 \end{gathered}$$

So,

.$$\begin{gathered} \sum _{n=0}^{\infty }{a}_n{x}^{n+1}=\sum _{k-1=0}^{\infty }{a}_{k-1}{x}^k \\ \sum _{n=0}^{\infty }{a}_n{x}^{n+1} =\sum _{k=1}^{\infty }{a}_{k-1}{x}^k \end{gathered}$$

Hence, the required term is $\sum _{k=1}^{\infty }{a}_{k-1}{x}^k$ .