Calculate Sums with F(i) Function for Math Problem

[daniel_i_l]

I need to calculate
$$\sum_{i=1}^n \frac{1}{i(i+1)}$$
useing the fact that:
$$\sum_{i=1}^n F(i) - F(i-1) = F(n) - F(0)$$
now I chose the function
$$F(i) = \frac{1}{i} \frac{1}{(i+1)} ... \frac{1}{(i+r)}$$
so
$$F(i)-F(i-1)=(\frac{1}{i}\frac{1}{(i+1)} ... \frac{1}{(i+r-1)})(\frac{1}{(i+r)}-\frac{1}{(i-1)})$$
now I want to use that to calculate the sum chooseing r as 2, but I'm stuck because the F(0) is undefined, and because of the
$$\frac{1}{i+r}-\frac{1}{i-1})$$

 

[shmoe]

You need to choose F(i) so that

$$F(i)-F(i-1)=\frac{1}{i(i+1)}$$

Your choice of F(i) does not satisfy this (what is that r anyways?) Try another choice of F, with a hint-think partial fractions.

 

[Physics Monkey]

Why on Earth would you choose that three term nightmare for F? There is a much easier way to do it. Hint: what is $$\frac{1}{i+1} - \frac{1}{i}$$?

 

[daniel_i_l]

Thanks a lot!