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bahrom7893
For what values of r>0 does the series converge?
I'm wary of using the integral test before actually getting some advices: |dw:1330831400158:dw|
.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.
@Mertsj can u take a look?
uh I got it from a calculator it says that its converge but I don't know how it solves, lol
@malevolence19 can u help?
sry for the disappointment :D
Okay, I think I can help you.
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 :/)
mal19 almost had it. If |r| < 1, then the limit of the ratio a_{n+1}/a_n converges to zero.
It's that simple? dang it and i'm sitting here with improper integrals!
*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.
namely, it converges to |r| itself.