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Chapter 7: Q. 32 (page 639)

In Exercises 29鈥�34 use the ratio test to analyze whether the given series converges or diverges. If the ratio test is inconclusive, use a different test to analyze the series

鈭�k=1鈭�k!(2k)!

Short Answer

Expert verified

The series converges.

Step by step solution

01

Step 1. Given information.

The given series is鈭�k=1鈭�k!(2k)!.

02

Step 2. Root test.

According to the series,

ak+1ak=(k+1)!(2k+2)!k!(2k)!=(2k)!(k+1)!k!(2k+2)!ak+1ak=(2k)!(k+1)k!k!(2k+2)(2k+1)(2k)!=(k+1)2(k+1)(2k+1)=12(2k+1)

03

Step 3. Conclusion.

On taking limits,

limk鈫�鈭�ak+1ak=limk鈫�鈭�12(2k+1)=12limk鈫�鈭�12k+1=0Since,L<1,

Therefore, the series converges.

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Most popular questions from this chapter

Use the divergence test to analyze the given series. Each answer should either be the series diverges or the divergence test fails, along with the reason for your answer.

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In Exercises 48鈥�51 find all values of p so that the series converges.

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