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Convergent sequences

Let cn be a sequence such that the subsequence ckn converges for each k=2,3,4,.... Need cn converge?

(edited 12 years ago)

What have you thought about so far? You should try writing out what the terms of each of c2n,c3n,c4n.

Edit: Don't do this, I was answering a different question

(edited 12 years ago)

Reply 2

DFranklin

Original post by IrrationalNumber

What have you thought about so far? You should try writing out what the terms of each of c2n,c3n,c4n.

I confess, I can't see how that will help (which is probably me being dim, but if I can't see it there's a good chance the OP can't either).

On the other hand, I'm somewhat struggling with how to give a useful hint that doesn't give the game away, so I'm happy to see what comes of your suggestion (or maybe you can make it a bit clearer).

Reply 3

DFranklin

{Having had a PM conversation with IrrationalNumber, it seems he was answering the original version of the post before it was edited. So I'll try to hint towards my solution}.

Hint 1: If you change a finite number of elements of a sequence, you don't affect whether or not it converges.

Hint 2: The simplest convergent sequence is a_n = 0.

Hint 3: So, we want to change some elements of a_n to 1, so that a_n doesn't converge.

Hint 4: But at the same time, we want each subsequence to converge, so we only want to change a finite number of elements of each subsequence a_kn.

Hint 5:

Hint 6:

Hint 7:

Reply 4

4321

Thank you so much! This is really helpful. Does that mean that the series with 0 as every element except 1 for cp where p is a prime would work? As only at most one member of each subsequence would be changed, but there are an infinite number of primes, so the series would not converge.
Thanks!

Reply 5

IrrationalNumber