Q- 26E - 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 .

26. $\sum _{n=1}^{\infty }\frac{a_n}{n+3}{x}^{n+3}$

Step-by-Step Solution

Answer

The required term is $\sum _{k=4}^{\infty }\frac{a_{k-3}}{k}{x}^k$

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 and c is a constant.

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

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, $f\left(x\right)=\sum _{n=1}^{\infty }\frac{a_n}{n+3}{x}^{n+3}$.

Let,

$$\begin{gathered} n+3=k \\ n = k-3 \end{gathered}$$

So,

$$\begin{gathered} \sum _{n=1}^{\infty }\frac{a_n}{n+3}{x}^{n+3}=\sum _{k-3=1}^{\infty }\frac{a_{k-3}}{k}{x}^k \\ =\sum _{k=4}^{\infty }\frac{a_{k-3}}{k}{x}^k \end{gathered}$$

Hence, the required term is $\sum _{k=4}^{\infty }\frac{a_{k-3}}{k}x$.