1. ## Summing up Series

Given the series:
1/((n+1)(3^(n+1))
Prove that it converges or diverges. If it converges, find its sum.

I can prove that it converges, but how would I find the sum?

2. ## Re: Summing up Series

You missed the most crucial information.

3. ## Re: Summing up Series

Ah, from 1 to infinity.

4. ## Re: Summing up Series

Well, just find the partial sums and then take the limit of it as it approaches infinity.

5. ## Re: Summing up Series

Figured it out. It wasn't partial sums.

-ln(1 - x) = sum from 0 to infinity of (x^(n+1))/(n + 1)
In my case, x = 1/3.
So, the sum is -ln(2/3).

6. ## Re: Summing up Series

It's -1/3 + ln(3/2)

7. ## Re: Summing up Series

My work:

What am I missing?

EDIT: It starts at 0, not 1 once I looked over the problem again.

8. ## Re: Summing up Series

Dat handwriting.

Anyway

If it's zero then it's ln(3/2) but:

-ln(2/3) = -ln(2) - (-ln(3)) = ln(3) - ln(2) = ln(3/2)

So it's the same thing.

9. ## Re: Summing up Series

Late to the party but (-1)*ln(2/3) = ln((2/3)^(-1)) = ln(3/2) is another way to show that.

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