What is a Closed Form for the Sequence?

[bonfire09]

Homework Statement

Let Ʃ 1\(n^2-1) from n=2 to k. It says find a closed for it and prove it using sum notation.

Homework Equations

The Attempt at a Solution

I can easily prove it by induction but I don't know what a closed form means. I tried looking it up online but there really isn't much info and nothing is stated in my textbook about. All I know is the sequence when expanded looks like

1/3+1/8+1/15+1/24+...+1/n^2-1. Then not sure how to put it in closed form?

 

[Dick]

Homework Statement

Let Ʃ 1\(n^2-1) from n=2 to k. It says find a closed for it and prove it using sum notation.

Homework Equations

The Attempt at a Solution

I can easily prove it by induction but I don't know what a closed form means. I tried looking it up online but there really isn't much info and nothing is stated in my textbook about. All I know is the sequence when expanded looks like

1/3+1/8+1/15+1/24+...+1/n^2-1. Then not sure how to put it in closed form?

Closed form means find a formula for the sum of series without writing out each term. Try to express 1/(n^2-1) in the form A/(n+1)+B/(n-1). Find A and B. Like partial fractions. Then start writing out terms of that and think about it.

 

[bonfire09]

This is what I get$$\left( \sum_{n=0}^k\frac{1}{-2(n+1)}+\frac{1}{2(n-1)} \right)=\frac{1}{3}+\frac{1}{8}+...+\frac{1}{-2(k+1)}+(\frac{1}{2(k-1)})$$. From here I am not sure how to get a closed sum.

 

[Dick]

This is what I get$$\left( \sum_{n=0}^k\frac{1}{-2(n+1)}+\frac{1}{2(n-1)} \right)=\frac{1}{3}+\frac{1}{8}+...+\frac{1}{-2(k+1)}+(\frac{1}{2(k-1)})$$. From here I am not sure how to get a closed sum.

Yes! It's (1/2)*(1/(n-1)-1/(n+1)). Start the sum at n=2 like your original problem posed and bring the (1/2) outside. The series then goes like 1/1-1/3+1/2-1/4+1/3-1/5+1/4-1/6+1/5-1/7... Don't you see a lot of cancellations in there?

 

[bonfire09]

Yes but all I'm left is 1+1\2?

 

[Dick]

Yes but all I'm left is 1+1\2?

If you are doing an infinite sum, yes. But if you dealing with the finite sum 2...k you also have some stuff that won't cancel at the k end of the sum.