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Chapter 8: Q 37. (page 670)

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.

鈭�k=0鈭�k!xk

Short Answer

Expert verified

The interval of convergence for power series is 0.

Step by step solution

01

Step 1. Given information.

The given power series is:

鈭�k=0鈭�k!xk

02

Step 2. Find the interval of convergence.

Let us assumebk=k!xk therefore

bk+1=k+1!xk+1

The ratio for the absolute convergence is

limk鈫�鈭�bk+1bk=limk鈫�鈭�k+1!xk+1k!xk=limk鈫�鈭�k+1x=limk鈫�鈭�xk+1

Here if we put role="math" localid="1649434511807" x=0, then the value of the limit will be turned to be zero and it does not matter what valuekhas. On the other hand if role="math" localid="1649434522725" x鈮�0then the value of the limit turns out to be infinite.

So, by the ratio test of absolute convergence, we know that series will converge absolutely when role="math" localid="1649434532723" x=0.


Therefore, the interval of convergence of the power series is0.

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Most popular questions from this chapter

How may we find the Maclaurin series for f(x)g(x) if we already know the Maclaurin series for the functions f(x) and g(x)? How do you find the interval of convergence for the new series?

What is a difference between a Taylor polynomial and the Taylor series for a function f at a point x0?

Why is it helpful to know the Maclaurin series for a few basic functions?

In Exercises 41鈥�48 find the fourth Taylor polynomial P4(x)for the specified function and the given value of x0.

48.sin3x,蟺6

Let 鈭�k=0鈭�鈥�akxkbe a power series in xwith a positive and finite radius of convergence p. Explain why the ratio test for absolute convergence will fail to determine the convergence of this power series when x=por when x=-p.
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