Q-24E - 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 .

24.$\sum _{n=2}^{\infty }n(n-1)a_n{x}^{n+2}$

Step-by-Step Solution

Answer

The required term is .$\sum _{k=4}^{\infty }(k-2)(k-3)a_{k-2}{x}^k$

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, represents the coefficient term of the nth term, is a constant.

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

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, $f\left(x\right)=\sum _{n=2}^{\infty }n(n-1)a_n{x}^{n+2}$ .

 

Let,

$$\begin{gathered} n+2=k \\ n=k - 2 \end{gathered}$$

So,

$$\begin{gathered} \sum _{n=2}^{\infty }n(n-1)a_n{x}^{n+2}= \sum _{k-2=2}^{\infty }(k-2)(k-2-1)a_{k-2}{x}^k \\ \sum _{n=2}^{\infty }n(n-1)a_n{x}^{n+2}= \sum _{k=4}^{\infty }(k-2)(k-2-1)a_{k-2}{x}^k \end{gathered}$$

Hence, the required term is .$\sum _{k=4}^{\infty }(k-2)(k-2-1)a_{k-2}{x}^k$.