Does #a_n=(5^n)/(1+(6^n) #converge? If so what is the limit?

1 Answer
Nov 10, 2015

Yes, it converges to zero.

Explanation:

First of all, I want to prove that #1+6^n# and #6^n# are asymptotically equivalent. To do so, we need to show that

#lim_{n\to\infty} (1+6^n)/6^n=1#

And a way to show it is factoring #6^n#:

#lim_{n\to\infty} (1+6^n)/6^n = lim_{n\to\infty} (cancel(6^n)(1+1/6^n))/cancel(6^n)= lim_{n\to\infty} 1+1/6^n = 1#

Since the two expressions are equivalent when #n->infty#, we can claim that

#lim_{n\to\infty} 5^n/(1+6^n) =lim_{n\to\infty} 5^n / 6^n = lim_{n\to\infty}(5/6)^n#

And now use the fact that #a_n = k^n# converges to zero if and only if #|k|<1#, which is our case.