## bahrom7893 3 years ago For what values of r>0 does the series converge?

1. bahrom7893

I'm wary of using the integral test before actually getting some advices: |dw:1330831400158:dw|

2. .Sam.

yeah

3. bahrom7893

.Sam. yeah what? Sorry to sound rude, but what's the point of disappointing me? I got so happy when I saw that notification pop up.

4. bahrom7893

@Mertsj can u take a look?

5. .Sam.

uh I got it from a calculator it says that its converge but I don't know how it solves, lol

6. bahrom7893

@malevolence19 can u help?

7. .Sam.

sry for the disappointment :D

8. bahrom7893

tnx for tryin sam

9. malevolence19

10. bahrom7893

awesome

11. malevolence19

For convergence of a series using the ratio test we must have that: $\lim_{n \rightarrow \infty}a_n \rightarrow 0$ And: $\lim_{n \rightarrow \infty} \left| \frac{a_{n+1}}{a_n}\right|<1$ So: $\lim_{n \rightarrow \infty} \left| \frac{r^{\ln(n+1)}}{r^{\ln(n)}}\right|=\lim_{n \rightarrow \infty} \left| r^{\ln(n+1)-\ln(n)}\right|=\lim_{n \rightarrow \infty} \left| r^{\ln \left( \frac{n+1}{n}\right)}\right| \rightarrow 1$ Means that the ratio test is inconclusive. We need to find another approach (I should have realized this wouldn't work, its a rational function :/)

12. bahrom7893

@JamesJ help?!

13. JamesJ

mal19 almost had it. If |r| < 1, then the limit of the ratio a_{n+1}/a_n converges to zero.

14. bahrom7893

oh so that's it?

15. bahrom7893

It's that simple? dang it and i'm sitting here with improper integrals!

16. JamesJ

*correction: If |r| < 1, then the limit of the ratio a_{n+1}/a_n converges to a number less than 1. And then by that ratio test, the sum converges.

17. bahrom7893

U guys rule!

18. JamesJ

namely, it converges to |r| itself.