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Use any convergence tests to determine whether the series converge absolutely, converge conditionally, or diverge. Explain why the series meets the hypotheses of the test you select.

k=1-1ke-k2

Short Answer

Expert verified

The series converges absolutely.

Step by step solution

01

Step 1. Given information.

Consider the given question,

k=1-1ke-k2

02

Step 2. Consider the general series.

The general term of the series k=1ak=k=1-1ke-k2is given by,

ak=-1ke-k2

The ratio role="math" localid="1649154069574" ak+1akis given by,

role="math" localid="1649154169585" ak+1ak=-1k+1e-k+12-1ke-k2=-1e-k+12e-k2=ek2e(k+1)2

03

Step 3. Find the value of limit.

The value of limkak+1akis given below,

role="math" localid="1649154467758" limkak+1ak=limkek2ek+12=limk1e2k+1=limk1e·e2k=0

The value of limkak+1akis 0, which is less than 1.

Thus, by ratio test for absolute convergence, the series k=1-1ke-k2 is absolutely convergent.

Hence, the given series is absolutely convergent.

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