Coalescence

Sum of the Infinite Series

[math]\sum_{i=2}^\infty\frac{2N}{\frac{i(i-1)}{2}}=\sum_{i=2}^\infty\frac{4N}{i(i-1)}=4N\sum_{i=2}^\infty\frac{1}{i(i-1)}[/math]

Note shifting the index starting point down by one, i=1 instead of i=2 in the sum.

[math]4N\sum_{i=2}^\infty\frac{1}{i(i-1)}=4N\sum_{i=1}^\infty\frac{1}{i(i+1)}=4N\sum_{i=1}^\infty\frac{1}{i}-\frac{1}{i+1}[/math]

Why is

[math]\frac{1}{i(i+1)}=\frac{1}{i}-\frac{1}{i+1}[/math]?

Multiply both sides by one to equalize the denominators and combine.

[math]\frac{1}{i}-\frac{1}{i+1}=\frac{i+1}{i+1}\frac{1}{i}-\frac{i}{i}\frac{1}{i+1}=\frac{}{i(i+1)}[/math]