00:01
In the question we are given series s as equal to the summation of x squared, sorry, n squared where it is n squared, n squared x to the power n, divided by where it is 6, 12, multiplied by 18, under the dash 6 n to the power 1.
00:25
So here it is 1.
00:27
So this is the summation from n equals to 1.
00:31
To infinity.
00:33
Let the means it is the x which is given is summation of n equal to 1 to infinity n squared by x to the power n divided by 6 multiplied by 12 multiplied by 18 and up to 6 now let's solve it.
00:52
So in the the limit will be seen as the condition in equal to 1 to infinity the numerator will be n square x to the power n.
01:01
But in the denominator, we can write 6 x x into 1, 12 x multiplied by 2, 18 is 6 multiplied by 3, and 6m 6 multiplied by n.
01:15
So 6 will be repeated n times, so this can be written as 6 to the power n, and in the bracket 1 multiplied by 2, multiplied by 3 up to n.
01:25
Okay, so looking it carefully, this term.
01:30
This term is nothing about n factorial and this is a product of n natural numbers which is n factorial only.
01:39
So this implies our s will be equal to summation of n equal to infinity x n square x to the power n divided by 6 to the power n into multiplied the n factorial.
01:56
So our a .n term will be here, n squared divided by 6 to the power n multiplied by n factorial, multiplied by x to the power n.
02:08
And the radius of convergence is nothing but limit and pending to infinity a n plus 1 divided by n.
02:19
So, this division is nothing but putting n plus 1 here instead of 1.
02:26
So we will get n plus 1 whole square x to be power n plus 1 divided by 6 to the power n plus 1 factorial.
02:40
Now this is the annette nf term we have to do added here.
02:45
So the this term will be 6 to be par on multiplied by n factorial will divide n square multiplied by x to the power n.
02:55
So x to the power n will be cancelled off and 1 x will be remaining here.
03:03
Similarly 6 to the power n will be canceling up and 1 6 will be remaining here.
03:09
Now n plus 1 square is n plus 1 multiplied the n plus 1.
03:14
So if this is equal to n plus 1 multiplied by n plus 1.
03:17
And n plus 1 hectare is nothing but n plus 1 multiplied by n factorial.
03:22
So 1 n plus 1 will be cut up here and this n factorial will be cancelled from here.
03:28
So this implies our r value will be equal to limit and tending to infinitive.
03:35
And the remnant terms will be n plus 1 divided by n square multiplied by 1 x was remaining here and 1 6 was remaining here...