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Chapter 1: Q-23E (page 1)

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

with generic term Xk.

23.鈭�n=1鈭�nanxn-1

Short Answer

Expert verified

The required term is .鈭�k=0鈭�(k+1)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 c, is a constant.

02

To express the given series in terms of 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=1鈭�nanxn-1

Let,

n-1=kn=k+1

So,

鈭�n=1鈭�nanxn-1=鈭�k+1=1鈭�(k+1)ak+1xk鈭�n=1鈭�nanxn-1=鈭�k=0鈭�(k+1)ak+1xk

Hence, the required term is鈭�k=0鈭�(k+1)ak+1xk

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