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Prove that if the series k=1akdiverges, then the series k=1akalso diverges.

Short Answer

Expert verified

The series k=1akis convergent

The seriesk=1akis divergent

Step by step solution

01

Step 1. Given information

k=1akandk=1akare the given series

02

Step 2. Finding whether ∑k=1∞ akis convergent

Assume that k=1akis not divergent,

Therefore, k=1akis convergent.

If k=1akis convergent but it is given that it is divergent.

If the series k=1akis absolutely convergent, then the series k=1akis convergent.

Therefore, k=1akis convergent, which is a contradiction as it is given that the series k=1akis divergent.

Therefore, the supposition that the series k=1akis not divergent is wrong.

Hence, the seriesk=1akis divergent.

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