Question

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

Answer 1

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.