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

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1. sam30317

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

2. Thomas9

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

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

4. Thomas9

only the 4 is raised to the power n.

5. sam30317

Nope, so is the 3.

6. Thomas9

in the first part I mean.

7. Thomas9

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

8. sam30317

Then what do i get??

9. Thomas9

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

10. sam30317

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

11. Thomas9

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

12. Thomas9

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

13. sam30317

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

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

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

16. sam30317

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

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

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

19. sam30317

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

20. Thomas9

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

21. Thomas9

Anyway, good luck.

22. sam30317

thaaanks!!

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