1. converge or diverge

does this series converge or diverge?

n = 2 E infinity

2/(n^2 -1)

..i've attempted to solve this, and i know the answer is 3/2

just not sure why i could not divide the numerator and denominator by the largest exponent in the denominator (n^2). is that because its not in the form of infininty/infinity?

also, tried using summation notation, it did not work out.

i know it can't be the harmonic series because that would diverge.

2. Originally Posted by rcmango
does this series converge or diverge?

n = 2 E infinity

2/(n^2 -1)
.
$\left| \frac{2}{n^2 - 1} \right| \leq \frac{2}{n^2 - \frac{1}{2}n^2} = \frac{4}{n^2}$
Now use comparasion test.

3. Originally Posted by rcmango
does this series converge or diverge?

n = 2 E infinity

2/(n^2 -1)

..i've attempted to solve this, and i know the answer is 3/2

just not sure why i could not divide the numerator and denominator by the largest exponent in the denominator (n^2). is that because its not in the form of infininty/infinity?

also, tried using summation notation, it did not work out.

i know it can't be the harmonic series because that would diverge.

to find the sum, you can realize that you have a telescoping sum. note that $\frac 2{n^2 - 1} = \frac 1{n - 1} - \frac 1{n + 1}$

write out some of the terms of the series (be sure to include the last few terms as well, that is, say, the (n - 2)th term, the (n - 1)th term, the nth term). try to see a pattern of what cancels out and come up with an expression for what's left. then let $n \to \infty$ and you should get your result

4. Okay, thanks for the help, i can't used the comparison test because i haven't learned it yet.

however i can use the telescoping series test, is that the same as the collapsing series?

also.. i've attached an image of what work i've done, I apoligize it is very messy, sorry was in a hurry.

btw, i see how the only terms of 1/2 and 1 are left only. but what happened to the (1/n-1) - (1/n+1) do those cancel out completely?

we used partial sums here to split the fraction up then?

thanks alot.

5. Originally Posted by rcmango
Okay, thanks for the help, i can't used the comparison test because i haven't learned it yet.

however i can use the telescoping series test, is that the same as the collapsing series?
i don't know. don't think i've heard the term "collapsing series" before.

also.. i've attached an image of what work i've done, I apoligize it is very messy, sorry was in a hurry.
where's the image?

btw, i see how the only terms of 1/2 and 1 are left only. but what happened to the (1/n-1) - (1/n+1) do those cancel out completely?
this is why i told you to write out some terms. you'd realize that the (n - 2)th term cancels the $\frac 1{n - 1}$, but the $\frac 1{n + 1}$ stays. you'd have to take the limit as $n \to \infty$ to get the answer

we used partial sums here to split the fraction up then?
i used partial fractions decomposition to get the two fractions