Question

summation from k=1 to infinity of (1/(5^k+1))

Answer 1

\[\sum_{1}^{\infty} \frac{ 1 }{ 5^k +1 }\]

Answer 2

Checking for convergence, or actually finding a sum?

Answer 3

Actually finding a sum

Answer 4

I can see why @SithsAndGiggles is having doubts with this question, because even if you just compute a few terms, you'll see that this converges, so I'd recommend you to see to which value it converges with the ratio test and take this as an answer, because the exact answer depends on how good your computer is (-:

Answer 5

Alright, thank you.

Answer 6

Really I'm just lookign for confirmation right now. I'd done out the problem and gotten 13/60 as an answer, which approximates to 0.216667, and on Wolfram Alpha the approximation is 0.215062.
I just wasn't sure if there is a more accurate answer to be found with just a graphing calculator.

Answer 7

I think this is as correct as you can get with it, I don't know of any formula to compute this by hand, so one way is to just sum up the few first terms and write it as a quotient, neglecting large denominators because they can be neglected:

\[\Large \sum_{n=1}^\infty\frac{1}{5^n+1}=\frac{1}{6}+\frac{1}{26}+\frac{1}{126}+\frac{1}{626}+...+0 \]

Answer 8

I had just used the common ratio with the sum of a geometric series = a/(1-r)
You can solve for the first two terms and find a and r via those first two.