Question

Have can I show this series are convergent or divergent?

Answer 1

Use the squeeze theorem and comparison tests

Answer 2

Wouldn't it be Leibniz's test?

Answer 3

I have never head about the squeeze theorem and neither the Leibniz's test. Can someone see another way to solve the series?

Answer 4

of course I know the comparison test..

Answer 5

just use the limit comparison test

Answer 6

Leibniz's test is the one for alternating series. Squeeze theorem is the one we compare with two other functions' limits, generally easier ones. And yeah, we don't use Leibniz's test here.

Answer 7

the series shall not be positive to use the limit comparison test?

Answer 8

again, just use the limit comparison test

Answer 9

\[\frac{\frac{n+(-1)^n7}{2n^2+1}}{\frac{1}{n}}\to\frac{1}{2}\text{ as }n\to\infty\]

Answer 10

yes, but the series should not be positive in order to use the limit comparison?

Answer 11

Or are I missing something?

Answer 12

it only matters that the series is eventually positive

Answer 13

just the simple comparison one. But isn't 1/2 inconclusive?

Answer 14

no...it tells us that the original series diverges

Answer 15

I mean, 0<=limt<=inf both can diverge and both can converge.

Answer 16

\[0<1/2<\infty\] and \[\sum_{n=1}^{\infty}\frac{1}{n}\] diverges

Answer 17

Oh, right... I see. I'm also learning series.

Answer 18

I just didn't see the previous posts. That's why I missed 1/n :P

Answer 19

Thank you @Zarkon