Question

Find the nature of the series using raabe-duhamel criteria or ratio criteria(D'Alembert)

Answer 1

\[\sum_{n=1}^{\infty}\ln(1+a^n),a \ge0\]

Answer 2

@ganeshie8

Answer 3

I think all both ratio test and raabe are inconclusive
you need to try something else

Answer 4

@eliassaab @SithsAndGiggles @dan815

Answer 5

so what are these criteria raabe-duhamel criteria or ratio criteria

Answer 6

i think there will be like a radius of convergence given some a

Answer 7

find that a

Answer 8

a<=1

Answer 9

but im pretty sure they want more than that, nature maybe HOW its converging and diverging as a function of a

Answer 10

for the sum itself u are dealing with a different a

Answer 11

like if a =1, u will have inf * ln (2)so diverging

Answer 12

u have to find the critical 'a'

Answer 13

@dan815 it's pretty clear that this exercise must be done using one of those criteria

Answer 14

i think it comes down to rate of decrease of a^n when a<1 and rate of increase of ln function

Answer 15

oh okay then can u post that criteria stuff here

Answer 16

http://en.wikipedia.org/wiki/Ratio_test
http://en.wikipedia.org/wiki/Convergence_tests#Raabe-Duhamel.27s_test

Answer 17

@dan815

Answer 18

why isn't it \(a<1\)?

Answer 19

ratio test seems to work, unless my algebra is bad

Answer 20

Yes the ratio test work, using L'Hospital's rule.
\[
\lim_{n->\infty}\frac {\ln(1+ a^{n+1})}{\ln(1+ a^{n)}}=\\
\lim_{n->\infty}\frac{a \left(a^n+1\right)}{a^{n+1}+1}=a
\]

Answer 21

Of course one has to suppose that a <1, since for \( a \ge 1 \), the series divereges

Answer 22

I'm going to post my full solvation of this series