1. Mathematical Induction

Use mathematical induction to prove that

n
Σ 2/r(r^2-1) = (1/2) - (1/n(n+1)) n= 2,3,4.......
r=2

i have done the basic proof and got to the point

k+1
Σ = (1/2) - (1/k(k+1)) + 2/(k+1)(k+1^2-1)
r=2

I think i need to get to (1/2) - 1/(k+1)(k+2) but im not sure how to get there, any help would be greatly appreciated!

2. Re: Mathematical Induction

Originally Posted by gurnster76
Use mathematical induction to prove that
n
Σ 2/r(r^2-1) = (1/2) - (1/n(n+1)) n= 2,3,4.......
r=2

i have done the basic proof and got to the point

k+1
Σ = (1/2) - (1/k(k+1)) + 2/(k+1)(k+1^2-1)
r=2
Work with
$\frac{1}{2}-\frac{1}{k(k+1)}+\frac{2}{(k+1)((k+1)^2-1)}=~?$

3. Re: Mathematical Induction

That's where im stuck.

4. Re: Mathematical Induction

Originally Posted by gurnster76
That's where im stuck.
$\frac{2}{(k+1)((k+1)^2-1)}=\frac{2}{(k+1)(k^2+2k)}=\frac{2}{k(k+1)(k+2)}$

5. Re: Mathematical Induction

but how do i get from

(1/2) - (1/k(k+1)) + (2/(k+1)(k+2)

to (1/2) - (1/(k+1)(k+2)