   Chapter 11.7, Problem 31E

Chapter
Section
Textbook Problem

# Test the series for convergence or divergence.31. ∑ k = 1 ∞ 5 k 3 k + 4 k

To determine

To test: Whether the series is convergence or divergence.

Explanation

Result used:

(1) Test for Divergence: If limnan does not exist or limnan0 , then the series n=1an is divergent.

(2) The geometric series n=1arn1 is convergent if |r|<1 and divergent if |r|>1 .

(3) If the series n=1an is convergent then limnan=0 .

Calculation:

Consider the given series k=1ak=k=15k3k+4k .

Obtain the limit an ,

limkak=limk[5k3k+4k]=limn14k(5k34kk+1)=limn((54)k(34)k+1)

Here, the series k=1(34)k is the geometric series with r=34

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