00:01
Here we have an infinite series, the sum from n equals 1 to infinity, of 12 over n times n plus 2.
00:07
And we want to find its sum.
00:08
To do that, we need to find a partial fraction decomposition and treat this like a telescoping series.
00:15
So i'm going to write this as something over n plus something over n plus 2.
00:21
Call that a and b.
00:23
I'm not going to write it there because i'm going to do that later.
00:25
So i'm going to get that if i multiply by the denominator, i have 12 is equal to n plus 2.
00:32
Times a plus n times b now if n is equal to zero that means that 12 is equal to 2a or a is equal to 6 so we know a let's just plug in a there and if i plug n equals negative 2 in here that's going to give us um 12 is equal to 0 minus 2b so b is equal to negative 6 perfect so now we can well let's just factor out of 6 here and we're left with 1 over n minus 1 over n plus 2.
01:06
Now, if you wrote this out as a sum from 1 to infinity, notice that we have six times, let's just ignore that 6.
01:15
Well, we have 1 over 1...