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bahrom7893Best ResponseYou've already chosen the best response.0
I'm wary of using the integral test before actually getting some advices: dw:1330831400158:dw
 2 years ago

bahrom7893Best ResponseYou've already chosen the best response.0
.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.
 2 years ago

bahrom7893Best ResponseYou've already chosen the best response.0
@Mertsj can u take a look?
 2 years ago

.Sam.Best ResponseYou've already chosen the best response.0
uh I got it from a calculator it says that its converge but I don't know how it solves, lol
 2 years ago

bahrom7893Best ResponseYou've already chosen the best response.0
@malevolence19 can u help?
 2 years ago

.Sam.Best ResponseYou've already chosen the best response.0
sry for the disappointment :D
 2 years ago

malevolence19Best ResponseYou've already chosen the best response.0
Okay, I think I can help you.
 2 years ago

malevolence19Best ResponseYou've already chosen the best response.0
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 :/)
 2 years ago

JamesJBest ResponseYou've already chosen the best response.2
mal19 almost had it. If r < 1, then the limit of the ratio a_{n+1}/a_n converges to zero.
 2 years ago

bahrom7893Best ResponseYou've already chosen the best response.0
It's that simple? dang it and i'm sitting here with improper integrals!
 2 years ago

JamesJBest ResponseYou've already chosen the best response.2
*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.
 2 years ago

JamesJBest ResponseYou've already chosen the best response.2
namely, it converges to r itself.
 2 years ago
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