Question

Question: In Problems 23–26, express the given power series as a series

with generic term X^k.

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

Short answer

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

Step-by-step solution

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.

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 .$\sum _{k=4}^{\infty }(k-2)(k-2-1)a_{k-2}{x}^k$.