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Chapter 1: Q-25E (page 1)

Question: In Problems 23鈥�26, express the given power series as a series with generic term XK.

25.鈭�n=0鈭�anxn+1

Short Answer

Expert verified

The required term is鈭�k=1鈭�ak-1xk.

Step by step solution

01

Power series

A power series is an infinite series of the form,

鈭�n=0鈭�an(x-c)n=a0+a1(x-c)+a2(x-c)2+.....

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

02

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

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

Given that, f(x)=鈭�n=0鈭�anxn+1

Let,

n+1=kn=k-1

So,

.鈭�n=0鈭�anxn+1=鈭�k-1=0鈭�ak-1xk鈭�n=0鈭�anxn+1=鈭�k=1鈭�ak-1xk

Hence, the required term is 鈭�k=1鈭�ak-1xk.

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