00:01
In this question, we are asked to find the interval of convergence of each of the given series.
00:06
In the first case, it's a geometric series, and the first few terms are 1 plus 7x, plus 7 squared x squared, plus 7 cubed x, and so on.
00:31
The first term of the series equals to 1, and the common ratio equals to 7x.
00:40
And a geometric series converges if the absolute value of the common ratio is less than 1.
00:49
So we want the absolute value of 7x to be less than 1.
00:55
Therefore, the absolute value of x must be less than 1 over 7.
01:00
And this means that the interval of convergence is from negative 1 over 7 to 1 over 7.
01:13
Note that i'm using parentheses because the endpoints are not included.
01:18
In the second case, the geometric series trick wouldn't work, and we'll have to use the ratio test.
01:29
By the ratio test, we need to calculate the limit of the absolute value of an plus 1 over a .n.
01:36
As n goes to infinity, where a .n is the general term of the series.
01:47
We can ignore the negative 1 to the nth because the absolute value sign cancels them.
01:52
So we are going to get the limit of n plus.
02:00
N plus 3 times x to the n plus 1, which is a n plus 1, divide by n plus 2 times x to the n, x to the n cancels...