Find the sum of the series: [1/(n^1.5) - 1/(n + 1)^1.5]

[MarkSA]

Hello,

Find the sum of the series: summation from n=1 to infinity of: [1/(n^1.5) - 1/(n + 1)^1.5]

The other portion of this problem involved plotting out 10 values of partial sums/sequences for a(sub n), and S(sub n), as well as drawing a graph. I've done this part already. Another is to determine by looking at the graph if a series is convergent or divergent. From the graph it's obvious that it's convergent. Now I need to find the sum of the series...

I'm not sure how to do this. I was able to find the sum of a geometric sequence earlier using a/1-r, but I don't think i've been given a formula to sum this type of problem. What technique would be used to do this? From the graph, the sum appears to approach 1 as n-> infinity. I need to show this though somehow, I think.

 

[galactus]

Actually, this isn't too bad if you see the telesoping sum.

You have \(\displaystyle \sum_{n=1}^{\infty}(\frac{1}{\sqrt{n}}-\frac{1}{\sqrt{n+1}})\)

\(\displaystyle (1-\frac{1}{\sqrt{2}})+(\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{3}})+(\frac{1}{\sqrt{3}}-\frac{1}{\sqrt{4}})+.........\)

See?. They all cancel one another out except the 1. So.............what do we get?.

 

[MarkSA]

Thanks, it looks like it sums to 1.

If you were to determine this using limits, how would one go about it? I think i'd need to know the expression for the sequence and then take the limit of that?
I'm not sure how to derive the expression for the sequence though. Example says it should be 1 - 1/[(n+1)^1.5] but i'm not sure how that was arrived at.