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Chapter 8: Q 22. (page 670)

Find the interval of convergence for power series:鈭�k=0鈭�-2kk!xk

Short Answer

Expert verified

The interval of convergence for power series isR.

Step by step solution

01

Step 1. Given information. 

The given power series is鈭�k=0鈭�-2kk!xk.

02

Step 2. Find the interval of convergence. 

Let us assume bk=-2kk!xkand role="math" localid="1649237026776" bk+1=-2k+1k+1!xk+1

Ratio for the absolute convergence is

limk鈫�鈭�bk+1bk=limk鈫�鈭�-2k+1k+1!xk+1-2kk!xklimk鈫�鈭�x-2k+1

Now, we evaluate the limit at k鈫�鈭�

So, limk鈫�鈭�x1k+1=0that is the value of limit will be zero no matter what value the variable xtakes.

By ratio test, the series converges absolutely for every value of x.

Therefore, the interval of convergence of the power series鈭�k=0鈭�-2kk!xkisR.

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Most popular questions from this chapter

Let ak鈮�0for each value of k, and let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a positive and finite radius of convergence p. What is the radius of convergence of the power series鈭�k=0鈭�鈥�1akxk?

Find the interval of convergence for power series:鈭�k=0鈭�k+13k2x-5k

Let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a finite radius of convergence p. Prove that if the series converges absolutely at either 卤p, then the series converges absolutely at the other value as well.

Show that the power series 鈭�k=1鈭�鈥�(鈭�1)kk2xkconverges absolutely when x=1and when x=-1. What does this behavior tell you about the interval of convergence for the series?

Explain why 鈭�k=0鈭�x-kk is not a power series.
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