00:01
In this question, we are asked to find the closed form of the following sum.
00:08
To do that, we will need the three formulas on the right -hand side.
00:13
And first, let's factor out 2.
00:17
From the square, from parenthesis, we will get 4 times 1 plus k over n squared, k from 1 to n.
00:31
Now this is equal to, since 4 doesn't depend on k, we can take it outside the sum.
00:38
We'll get some of...
00:40
Now let's expand the square.
00:43
We will get 1 plus 2 times k over n plus k squared over n squared.
00:56
Now we can split this sum into 3 sums.
01:02
We'll get 4 multiplied by sum of 1s from 1 to n plus 2 times sum k over n.
01:14
K from 1 to n plus the sum of k squared over n squared now note that 1 over n in k over in the sum k over n 1 over n doesn't depend on k so we can also take it outside the sum we're going to get sum of ones plus 2 over n times sum of k's plus 1 over n times sum of k's plus 1 over n squared times sum of k squared.
02:04
Now we are going to use the formulas at the beginning.
02:11
So the sum of once is equal to 2n plus 2 over n.
02:20
Now the sum of k is equal to n times n plus 1 over 2...