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Hi, I have this Infinite Series problem, and I need help with it. It's attached to the question. Thank you The question is, to find the sum of...

We are asked to find the sum `sum_(n=1)^(infty)(1/sqrt(n)-1/sqrt(n+1)) ` :


This is an example of a telescoping series; virtually all of the terms in any partial sum are eliminated. Consider the first few partial sums:


n=1: 1-1/2=1/2


n=2: (1-1/2)+(1/2-1/3)= 1 + (-1/2+1/2)-1/3=2/3


n=3: (1-1/2)+(1/2-1/3)+(1/3-1/4)=1+(-1/2+1/2)+(-1/3+1/3)-1/4=3/4


etc..


The mth partial sum is `1-1/sqrt(m+1) `  and it is obvious that as m tends to infinity the sum is 1.


(You have to be very careful -- not all telescoping sums converge.  Consider 1-1+1-1+1-1+1-1+... Grouped one way you get 1: 1+(-1+1)+(-1+1)+... but grouped another way you get 0: (1-1)+(1-1)+(1-1)+... Some telescoping series are divergent. Again, be careful.)


`sum_(n=1)^(infty)(1/sqrt(n)-1/sqrt(n+1))=1 `

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