Asked by Philip Martinson
Prove that the sequence: {an} = {(1 + (ln(6)/(n)))^(2n)}infinity n=1 converges
Note:
I don't know how to solve or work out so show all your work. And give the answer in EXACT FORM example 3pi, sqrt(2), ln(2) not decimal approximations like 9.424,1.4242,1232
Note:
I don't know how to solve or work out so show all your work. And give the answer in EXACT FORM example 3pi, sqrt(2), ln(2) not decimal approximations like 9.424,1.4242,1232
Answers
Answered by
oobleck
You know that (1 + 1/n)^n converges to e
You can easily verify that (1 + a/n)^n and (1+1/n)^(bn) converge.
google can provide proofs.
(1 + a/n)^n = e^a
(1 + 1/n)^(bn) → e^b
So now just combine them. (1 + a/n)^(bn) → e^(ab)
In this case, since e^ln6 = 6, that would be 6^2 = 36
You can easily verify that (1 + a/n)^n and (1+1/n)^(bn) converge.
google can provide proofs.
(1 + a/n)^n = e^a
(1 + 1/n)^(bn) → e^b
So now just combine them. (1 + a/n)^(bn) → e^(ab)
In this case, since e^ln6 = 6, that would be 6^2 = 36