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Use either the divergence test or the integral test to determine whether the series in Exercises 32–43 converge or diverge. Explain why the series meets the hypotheses of the test you select.

36.k=1kk2+3

Short Answer

Expert verified

The series is divergent.

Step by step solution

01

Step 1. Given information

We have been given the seriesk=1kk2+3

We have to determine whether the series converge or diverge.

02

Step 2. Determine whether the series converge or diverge.

Consider function fx=x3+x2

The function is continuous, decreasing, with positive terms.

All the conditions of integral test are fulfilled.

So, integral test is applicable.

Consider the integral localid="1649088344379" x=1fxdx=x=1x3+x2dx

localid="1649088398964" x=1fxdx=limkx=1kx3+x2dx=12limku=4k2+3duu(Put3+x2=u2xdx=du)=12limklnu4k2+3=12limklnk2+3-ln4=

The integral diverges.

So, the series is divergent.

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