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Given a series k=1ak, in general the divergence test is inconclusive when . For a ak0geometric series, however, if the limit of the terms of the series is zero, the series converges. Explain why.

Short Answer

Expert verified

Geometric series are convergent only when the ratio holds r<1.

Step by step solution

01

Step 1. Given Information.

The series:

k=1akak0

02

Step 2. Consider the series.

Consider the geometric series,

k=1cr4

The value of the series is zero only when the ratio is less than one.

03

Step 3. Geometric series are convergent.

Geometric series are convergent only when the ratio holds r<1.

So, for a geometric series, if the limit of the terms of the series is zero, the series converges.

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