00:01
Given this series, we're asked to find the values of x for which it converges, and then to find the sum of that series for those values of x.
00:08
And to do that, i'm going to use this fact, because this series right here, it's a geometric series, and actually i can rewrite it this way.
00:18
So still from n is equal to 0 to infinity, but now i'm going to write it as cosine of x divided by 8, that whole thing raised to the nth power.
00:32
So now it's obvious that this series is of this form, and therefore this series converges only when the absolute value of this argument is less than 1.
00:45
So the question is, when is the absolute value of cosine of x divided by 8 less than 1? and it turns out that that's always true.
00:56
So to see that, note that for all real values of x, so basically for x between minus infinity and infinity, we have that the absolute value of cosine of x, this is less than or equal to 1, and then i can take both sides of this inequality, divide through by 8, and then 1 8th, this is strictly less than 1.
01:26
Therefore, i have that this quantity is less than 1.
01:30
So let me write that out.
01:31
I have that cosine of x over 8, the absolute value of this is strictly less than 1...