00:01
Question i'm going to determine whether the series is convergent or divergent by expressing this as a telescoping sum.
00:06
If it's convergent i'm going to find its sum.
00:09
Here we have the sum from n equals 1 to infinity of 21 over n times the quantity of n plus 3.
00:17
So we're going to do a partial fraction decomposition on this.
00:22
We're gonna say okay 21 over n times the quantity of n plus 3 can be written in the form of a over n plus b over n plus 3.
00:37
So what are we gonna do? we're gonna multiply through by a least common denominator n times the quantity of n plus 3.
00:48
And if we do what do we get? well up top we have 21 and on the right hand side i'm getting a times the quantity of n plus 3 plus b times n.
01:02
Now to solve for a and b we're gonna choose values of n that allow us to do that.
01:09
Specifically i'm going to let n equals 0 and i'm going to let n equal negative 3.
01:16
So if n equals 0 this equation simplifies to 21 equals 3a so that a itself is equal to 7.
01:26
If i let n equal negative 3 i get 21 equals negative 3b so that b itself is negative 7.
01:37
So we substitute in and what do we have? we have a sum from n equals 1 to infinity of we've got a over n 7 over n and then it's b over n plus 3 minus 7 over n plus 3.
02:00
Now write out the first several terms of the series...