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Chapter 8: Q. 6 (page 692)

Explain why limn鈫�鈭�xnn!=0for every value of x.

Short Answer

Expert verified

Using ratio test of convergence,

limn鈫�鈭�xn+1n+1!脳n!xn=limn鈫�鈭�xnn+1=0

Step by step solution

01

Step 1. Given information.

The limit islimn鈫�鈭�xnn!=0.

02

Step 2. Explanation.

Use the ratio test of convergence.

Let bn=xnn!and bk+1=xn+1n+1!

role="math" localid="1649610457627" limn鈫�鈭�xn+1n+1!脳n!xn=limn鈫�鈭�xnn+1=0

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Most popular questions from this chapter

Fill in the blanks: The graph of every odd function is symmetric about ______. The graph of every even function is symmetric about ______.

What is Lagrange鈥檚 form for the remainder? Why is Lagrange鈥檚 form usually more useful for analyzing the remainder than the definition of the remainder or the integral provided by Taylor theorem?

Find the interval of convergence for power series:鈭�k=0鈭�-1k2k!x2k.

Show that the series:ln2+鈭�k=1鈭�-1k-1k2kx-2k

from Example 3 diverges when x = 0 and converges conditionally when x = 4.

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�13k+5kx+3k
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