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# Analysis question (sequences) watch

1. https://www8.imperial.ac.uk/content/...1P1/m1p1s2.pdf

I need help with the proof for 2(i). Clearly the limit is 0. But we can't explicitly solve |a_n| < e for n when a_n is what we have. So we need another simple sequence b_n such that |a_n| < |b_n| for all n sufficiently large and then we can solve |b_n| < e for n. But what can we choose for b_n? The first thought would be n^2/n^3 = 1/n. But this is in fact smaller than a_n (for n larger than 2) and so it doesn't seem to be of any use. So what can we use for our b_n? Is there some neat factorization?
2. Can we not take for say, n>1, which we can get into a multiple of 1/n and everything is hunky dorey?
3. (Original post by gangsta316)
what can we choose for b_n? The first thought would be n^2/n^3 = 1/n. But this is in fact smaller than a_n (for n larger than 2) and so it doesn't seem to be of any use. So what can we use for our b_n? Is there some neat factorization?
Your second thought should be 2n^2/n^3 = 2/n...
4. (Original post by DFranklin)
Your second thought should be 2n^2/n^3 = 2/n...
Ok. Are we absolutely sure that this is larger than a_n? We've made the top and bottom of the fraction bigger so we have to be sure that the bigger top has a greater effect.
5. (Original post by gangsta316)
Ok. Are we absolutely sure that this is larger than a_n?
Well, obviously you're not completely sure, so you'd better convince yourself. (Depending on how you do it, you might end up having a third thought).
6. (Original post by DFranklin)
Well, obviously you're not completely sure, so you'd better convince yourself. (Depending on how you do it, you might end up having a third thought).
How? Induction? I had another thought and even if I doubt 2/n surely something like 100000/n would definitely be bigger and would work?
7. (Original post by gangsta316)
How? Induction? I had another thought and even if I doubt 2/n surely something like 100000/n would definitely be bigger and would work?
So what's your problem? Simplest way of showing something works is to go ahead and do it.

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