Which of the following is not correct for a positive term series:a)Ifa converges, then an→0b)Ifconverges anddiverges, then divergesc)If an→0, thenconvergesd)The seriesconverges if r

Question

Which of the following is not correct for a positive term series:a)Ifa converges, then an&rarr;0b)Ifconverges anddiverges, then divergesc)If an&rarr;0, thenconvergesd)The seriesconverges if r<1.Correct answer is option 'C'. Can you explain this answer?

Answer

For (A), Let (S_n) be the sequence of partial sum of

.

Then

Since

converges, hence its sequence of partial sum is convergent.

Let

For (B), Let (S_n) and (t_n) are the sequence of partial sums of

, and

respectively. Then

be the sequence of partial sum of

.

Since

, converges, hence (s_n) is convergent. Also

diverges, so (t_n) is divergent.

then (s_n + t_n) is divergent, henve

diverges.

But by the statement of p-series test

diverges.

For (d), Note that an infinite geometric series

converges only when |r|<1.

hence statement (c) is incorrect

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