Q-23E - 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 .

23.$\sum _{n=1}^{\infty }n{a}_n{x}^{n-1}$

Step-by-Step Solution

Answer

The required term is .$\sum _{k=0}^{\infty }(k+1)a_{k+1}{x}^k$.

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 c, is a constant.

Step 2: To express the given series in terms of 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,

$f\left(x\right)=\sum _{n=1}^{\infty }n{a}_n{x}^{n-1}$

Let,

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

So,

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

Hence, the required term is $\sum _{k=0}^{\infty }(k+1)a_{k+1}{x}^k$