00:01
In this question, we are given the function f of x defined by a power series x to the n over n factorial.
00:11
First, let's write down the first few terms of the series.
00:16
The first term is going to be one.
00:19
The second term is x.
00:21
The third term is going to be x squared over 2 factorial, but that's same as x squared over 2.
00:27
Next term is x cube over 3 factorial plus so on.
00:33
Now, we are asked to answer several questions about this series, and the first question asks us to find the value of f at x equals 0.
00:45
Note that if we plug in x equal 0, everything cancels except for the first term.
00:53
Therefore, f of 0 is equal to 1.
00:58
In the next question, we are asked to find the domain of f.
01:06
And the domain of f is basically the set of values of x permissible for the series.
01:16
In other words, this is a set of values of x for which the series converges.
01:22
So basically, we are asked to find the interval of convergence of the series.
01:32
To do that, we are going to use the ratio test.
01:39
By the ratio test, we first need to find the limit of the absolute value of an plus 1 over an.
01:47
As n goes to infinity...