Question

Test convergence of series

Answer 1

so i know that this converges, because the a^k + 1 is always bigger than 0 and it is also bigger than one, which means the fraction is always below 1...but i am not sure how to solve it :/

Answer 2

What happens when a=1 ?

Answer 3

i have k on 1 +1 and for k=1 that's 1/2

Answer 4

we get \[\frac12+\frac12+\frac12 +... =\infty\]

the series diverges at a=1

Answer 5

doesn't diverging mean when the series is bigger than 1 ?

Answer 6

because like this no matter what a is the series will always be 1/m + 1/m + 1/m....

Answer 7

divergent means not convergent.

convergent series means when add up the terms , your answer converges on a specific number

Answer 8

at a =1 if you keep adding 1/2 you will get to infinity not a specific number. your series diverges at a=1

Answer 9

Also check what happens when 0<a<1 and when a > 1.

Answer 10

You can use the ratio test for convergence.

https://en.wikipedia.org/wiki/Convergent_series#Convergence_tests

Answer 11

okay, but is there a way to calculate this?
like some method?
or do i only explain it?

Answer 12

the ratio test isn't much helpful here, since i get