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# Sequence/Series Questions Watch

1. If An is a real sequence....

If n^2.(An) tends to zero as n tends to infinity, does the series Sum[An] converge?

If n.(An) tends to zero as n tends to infinity, does the series Sum[An] converge?

I'm pretty sure that the answer is yes in both cases, but constructing a nice understandable proof is proving to be more difficult than it should be.

Any hints/ideas would be much appreciated!
2. In the first case, can you use the comparison test to compare to 1/n^2?

In the second case, your answer is wrong. Can you use my hint for the first one to help you construct a counter example. (It diverges very slowly...)
3. Well a very famous series beginning with h can be used to counteract your yes reply to the second statement
4. Cheers I've got the first one out.

For the 2nd one am I being completely retarded when I say the 'n's are the same so if An= (1/n) n.(An) = n.(1/n) = 1 for all n, so won't tend to zero....? The harmonic series had crossed my mind about a hundred times but I discounted it for that reason.
5. (Original post by Hughh)
Cheers I've got the first one out.

For the 2nd one am I being completely retarded when I say the 'n's are the same so if An= (1/n) n.(An) = n.(1/n) = 1 for all n, so won't tend to zero....? The harmonic series had crossed my mind about a hundred times but I discounted it for that reason.
The harmonic series doesn't work. You need something that goes to 0 slightly faster than harmonic. But (1/n)^a will not work for a>1. You'll need something a bit slower than a power of n. Can you think of something?

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Updated: November 30, 2010
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