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Chapter 8: Q 49. (page 670)

Find the radius of convergence for the given series:鈭�k=0鈭�k!k+m!xk

Short Answer

Expert verified

The radius of convergence for the series is 1.

Step by step solution

01

Step 1. Given information. 

The given power series is鈭�k=0鈭�k!k+m!x.

02

Step 2. Find the radius of convergence.

Let us take bk=k!k+m!xktherefore bk+1=k+1!k+1+m!xk+1

Ratio for the absolute convergence is

limk鈫�鈭�bk+1bk=limk鈫�鈭�k+1!k+1+m!xk+1k!k+m!xk=limk鈫�鈭�k+1k+1+mx

So, by the ratio test of absolute convergence, the series will converge when x<1.

This implies that x鈭�(-1,1).

Therefore, the radius of the convergence for the series is1.

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Most popular questions from this chapter

What is meant by the interval of convergence for a power series in x? How is the interval of convergence determined? If a power series in xhas a nontrivial interval of convergence, what types of intervals are possible?

Find the interval of convergence for each power series in Exercises 21鈥�48. If the interval of convergence is finite, be sure to analyze the convergence at the endpoints.

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Let fbe a function with an nth-order derivative at a point xoand let Pn(x)=鈭�k=0nf(k)x0k!x-x0k. Prove thatf(k)x0=Pn(k)x0 for every non-negative integerk鈮�n.

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role="math" localid="1649406173168" 49.cosx,蟺2

Prove that if (-p,p]is the interval of convergence for the series 鈭�k=0鈭�鈥�akxk, then the series converges conditionally at p.

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