Question

determine whether series converges

sum from n=1 to infinity 1/(2n+3)^4

I know it converges but can anyone guide me through the test in which I approach that?

Why do I apply the integral test?

Answer 1

if the ratio and other test are inconclusive; the integral test is a valid option

Answer 2

since an integration is a larger sum than a discrete summation; if the larger sum converges, the lesser has to

Answer 3

how would i know the other tests don't work? There are so many. (ratio, p-series, geometric, comparison, etc

Answer 4

its not a matter of "doesnt work" its a matter of if they come up inconclusive; which I believe has to do with a zero

Answer 5

Also, maybe a comparison test? Compare with 1/(2n)^4, I think it's possible, or not?

Answer 6

for example;

1/n limits to zero; but it diverges; so the limit test is inconclusive if the limit = 0

Answer 7

ah. So how would I work out the problem? replace n's for x's, then use u-substitution u=the inside 2x+3? what do i do with the power 4? Can you carry it out for me?

Answer 8

so there's more than one way to get the answer?

Answer 9

if you ned to usub it thats fine ...

Answer 10

Generally, yes. There are multiple ways :-)

Answer 11

this whole series and sequence is killing me

Answer 12

it can be confusing since we have to think in terms of infinity itself

Answer 13

is there a general rule for each test?

Answer 14

yes, they tend to be printed in your textbooks .... i cant recall them at the moment tho

Answer 15

By the way, can we compare with just 1/n^4? Then, it converges because it's a p-series with p > 1. As 1/n^4 >= 1/(2n+3)^4, the original series converges.

Answer 16

And if you are using Stewart's Calc textbook, there's a cheat sheet on section 11.6, IIRC.

Answer 17

maybe 1/(2n)^4

Answer 18

the strategies for series, i know but it doesn't seem to help. I was hoping you guys have a trick that can help you remember :)

Answer 19

does patrickjmt videos help?

Answer 20

Think so. I thought they were useful at least. But I watched mostly the Taylors series videos from him

Answer 21

ok, ill refer to them but I'm just going to hit the books and see what I can do with the example problems.

Answer 22

Thanks

Answer 23

No problem. But I think in order to be good at series/sequences you have to do a ton of exercises. It's like integration, you have to acquire some experience. :-)