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Use any convergence test from this section or the previous section to determine whether the series in Exercises 31–48 converge or diverge. Explain how the series meets the hypotheses of the test you select.

k=1kk2+3

Short Answer

Expert verified

The series k=1kk2+3is convergent.

Step by step solution

01

Step 1. Given information.

The given series is the following.

k=1ak=k=1kk2+3

02

Step 2. The Limit Comparison Test. 

Consider a series k=1bk=k=11k32by taking the dominant term of numerator and denominator of k=1kk2+3.

Find the value of limkakbk.

limkakbk=limkkk2+31k32limkakbk=limkk12k32k2+3limkakbk=limkk2k2+3limkakbk=11+3k2limkakbk=1

Since k=1bk=k=11k32is convergent by the p-series test so k=1ak=k=1kk2+3is also convergent.

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