Question about series and telescoping sum

[vande060]

Homework Statement

∞, n=2 ∑ 2/(n² -1)

determine whether the series is divergent, convergent by expressing S_n as a telescoping sum. If it is convergent, find its sum

Homework Equations

The Attempt at a Solution

∞, n=2 ∑ 2/(n² -1)

okedokee, I broke the expression down by partial sums

∞, n=2 ∑ ( 1/(n-1) - 1/(n+1))

I did the first few expressions in the series, but I am not sure where to go after this

(1/1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + (1/4 - 1/6)...

i know that 1/3 should cancel, along with 1/4.

 

[Dick]

... is (1/5-1/7)+(1/6-1/8)+(1/7-1/9) then another ... So I think 1/5, 1/6 and 1/7 also cancel. Can't you extrapolate to all of the terms that cancel without writing ALL of them out?

 

[vande060]

... is (1/5-1/7)+(1/6-1/8)+(1/7-1/9) then another ... So I think 1/5, 1/6 and 1/7 also cancel. Can't you extrapolate to all of the terms that cancel without writing ALL of them out?

Well, after looking again I think that 1 + 1/2 will never cancel, and i have to add the nth term so

1 + 1/2 + 1/(n-1) - 1/(n+1), except the 1/(n-1) term can be taken out because it will be canceled

so the expression is

1 + 1/2 - 1/(n+1), is that right?

 

[Dick]

Well, after looking again I think that 1 + 1/2 will never cancel, and i have to add the nth term so

1 + 1/2 + 1/(n-1) - 1/(n+1), except the 1/(n-1) term can be taken out because it will be canceled

so the expression is

1 + 1/2 - 1/(n+1), is that right?

The upper limit isn't n. It's infinity. What's happens to your leftover term then?

 

[vande060]

The upper limit isn't n. It's infinity. What's happens to your leftover term then?

oh ok, so if n goes to infinity, that last term will become zero. so the sum will converge on 3/2 right?

thanks so much for your help so far

 

[Dick]

oh ok, so if n goes to infinity, that last term will become zero. so the sum will converge on 3/2 right?

thanks so much for your help so far

Right.