Sequence Convergence: Determine if an= (13(4n)+11)/(10(5n))

[nvalia]

Homework Statement

Determine if the sequence a_n converges.

a_n = (13(4ⁿ)+11)/(10(5ⁿ))

Homework Equations

N/A

The Attempt at a Solution

$$\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)})$$
= $$1/10 \lim_{n \to \infty} (\frac{(3(4^n))+11}{5^n})$$
= $$1/10 \lim_{n \to \infty} (\frac{(3(4^n)}{5^{n}}) + \lim_{n \to \infty} (\frac{11}{5^n})$$

I feel like I am missing something very basic. Thank you for your help!

 

[lanedance]

Homework Statement

Determine if the sequence a_n converges.

a_n = (13(4ⁿ)+11)/(10(5ⁿ))

Homework Equations

N/A

The Attempt at a Solution

$$\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)})$$
= $$1/10 \lim_{n \to \infty} (\frac{(3(4^n))+11}{5^n})$$
= $$1/10 \lim_{n \to \infty} (\frac{52^{n}}{5^{n}}) + \lim_{n \to \infty} (\frac{11}{5^n})$$

I feel like I am missing something very basic. Thank you for your help!

how did you get your last line? where did 52 come from?

what is $$\lim_{n \to \infty} (\frac{4}{5})^n$$?

 

[nvalia]

Oops! That was a typo -- fixed. Sorry about that.

And, if I write out the sequence, I would have to say the limit of (4/5)^n = 0.
Am I allowed to split the limit like I did, and, if so, are the limits separately 0 and 0, so that the final answer is "converges to zero"?

 

[lanedance]

it depends what has been proved, but where the limie exists and is finite for both f & g, generally it is fine to assume
$$\lim_{n \to \infty} (f(n) +g(n)) = \lim_{n \to \infty} f(n) +\lim_{n \to \infty}g(n)[/$$

$$\lim_{n \to \infty} (\frac{(3(4^n))+11}{10(5^n)}) = \lim_{n \to \infty} \frac{3}{10}(\frac{4}{5})^n + \lim_{n \to \infty} \frac{11}{10}(\frac{1}{5})^n$$