## sam30317 Group Title Does the series converge or diverge? Summation (n+3^n)/4^n n=0 to inf. 2 years ago 2 years ago

1. sam30317 Group Title

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

2. Thomas9 Group Title

I would seperate it in two parts: n/4^n+(3/4)^n Then I'd try the root test on both parts seperately.

3. sam30317 Group Title

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

4. Thomas9 Group Title

only the 4 is raised to the power n.

5. sam30317 Group Title

Nope, so is the 3.

6. Thomas9 Group Title

in the first part I mean.

7. Thomas9 Group Title

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

8. sam30317 Group Title

Then what do i get??

9. Thomas9 Group Title

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

10. sam30317 Group Title

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

11. Thomas9 Group Title

Well, $lim_{n \rightarrow \infty} \sqrt[n]{n} =1$ It's standard thing, good idea to remember that limit.

12. Thomas9 Group Title

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

13. sam30317 Group Title

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??

14. Thomas9 Group Title

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.

15. Thomas9 Group Title

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

16. sam30317 Group Title

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

17. Thomas9 Group Title

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.

18. Thomas9 Group Title

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

19. sam30317 Group Title

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

20. Thomas9 Group Title

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

21. Thomas9 Group Title

Anyway, good luck.

22. sam30317 Group Title

thaaanks!!