Question

Does the series converge or diverge? Summation (n+3^n)/4^n n=0 to inf.

Answer 1

\[\sum_{n=0}^{\inf} \frac{n+3^n}{4^n}\]

Answer 2

I would seperate it in two parts: n/4^n+(3/4)^n

Then I'd try the root test on both parts seperately.

Answer 3

So i would be left with (n/4) = infinity= div
(3/4) <1 = conv..
Because one div, they both dierge???

Answer 4

only the 4 is raised to the power n.

Answer 5

Nope, so is the 3.

Answer 6

in the first part I mean.

Answer 7

So you don't get n/4 when you do the root test.

Answer 8

Then what do i get??

Answer 9

\[\frac{\sqrt[n]{n}}{4}\]

Answer 10

lol, that looks more complicated. How would i solve that??

Answer 11

Well, \[lim_{n \rightarrow \infty} \sqrt[n]{n} =1\]

It's standard thing, good idea to remember that limit.

Answer 12

So \[\frac{\sqrt[n]{n}}{4} \rightarrow \frac{1}{4}\]
So that part does converge.

Answer 13

Oooo kaaay, lol. Thanks!! Im having a hard time remembering when all the tests conv or diverge. Like with n term test: if it equals 0, its inconclusive. And p series: p>1, conv.

How did u manage to memorize them w/o confusing yourself??

Answer 14

I only know the ratio and the root test, not sure what n term test or p series is.

For ratio and root test it's the same, <1 to converge, 1 inconlusive, so that's not too hard to remember.

Answer 15

I do know the n term test actually, thanks wiki.

Answer 16

Lol. I had a test on this stuff already and it was crazy how confused I got (i.e I failed it). Now my final is today and im going crazy, lol

Answer 17

Well the term test makes a lot of sense, if the limit doesn't equal zero, you're adding an infinite amount of numbers that aren't close to zero, so that's sure to diverge.

Answer 18

It's the first one to try, because it's easy to calculate.

Answer 19

Pseries, nth term, geom are easy to remember. I know about the root/ration , thanks to you! Now....on to others

Answer 20

general rule: if you see powers of n: use the ratio test,

if you see factorials: ratio test.

Answer 21

Anyway, good luck.

Answer 22

thaaanks!!