The question is: Prove that the sequence is monotone and bounded. Then find the limit. =1 and = ( +5)for n N. { } is increasing. Are we guessing that ?
Last edited by CountingPenguins; Jun 29th 2011 at 09:49 PM. Reason: Figured out the formatting.
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You need to use tex tags, not math. And your close tag needs /, not \.
Thanks for the pointer on the formatting. And I think the Lady Gaga equation at the bottom is HILARIOUS!
Originally Posted by CountingPenguins The question is: Prove that the sequence is monotone and bounded. Then find the limit. =1 and = ( +5)for n N. { } is increasing. Are we guessing that ? is monotone increasing: Induction on n: Say holds for : Proving for : . . . is bounded: Induction on : Say that for all is true. Proving for : So, is monotonic increasing and bounded, hence exists. We denote where .
Originally Posted by CountingPenguins The question is: Prove that the sequence is monotone and bounded. Then find the limit. =1 and = ( +5)for n N. { } is increasing. Are we guessing that ? Another solution: , Define new sequence: : is geometric sequence with and . Hence, . So,
Originally Posted by CountingPenguins The question is: Prove that the sequence is monotone and bounded. Then find the limit. =1 and = ( +5)for n N. { } is increasing. Are we guessing that ? First observe that if this converges it converges to a root of: knowing this greatly simplifies your work (in this case it provides a better guess for the upper bound for one thing). CB
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