bahrom7893 Group Title For what values of r>0 does the series converge? 2 years ago 2 years ago

1. bahrom7893 Group Title

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

2. .Sam. Group Title

yeah

3. bahrom7893 Group Title

.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 Group Title

@Mertsj can u take a look?

5. .Sam. Group Title

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

6. bahrom7893 Group Title

@malevolence19 can u help?

7. .Sam. Group Title

sry for the disappointment :D

8. bahrom7893 Group Title

tnx for tryin sam

9. malevolence19 Group Title

10. bahrom7893 Group Title

awesome

11. malevolence19 Group Title

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 Group Title

@JamesJ help?!

13. JamesJ Group Title

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

14. bahrom7893 Group Title

oh so that's it?

15. bahrom7893 Group Title

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

16. JamesJ Group Title

*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 Group Title

U guys rule!

18. JamesJ Group Title

namely, it converges to |r| itself.