00:01
In this question, we are asked to find the power series representation of the function f centered at 0, and then find the interval of convergence.
00:08
And to do that, we are going to use the formula 1 over 1 minus u equals to the series u to the n, n from 0 to infinity, which converges for the absolute value of u less than 1.
00:25
Now, we will rewrite the function f as 1.
00:30
First we are going to factor out 3 in the denominator to get 1 over 3 times 1 plus x over 3.
00:40
And the next step would be to rewrite it as 1 3 multiplied by 1 over 1 minus negative x over 3.
00:54
Note that i haven't changed anything.
00:57
However, now we can use the formula above by replacing u by negative x over 3.
01:07
We are going to get 1 third multiplied by the series negative x over 3 to the n and from 0 to infinity.
01:19
This equals to 1 third multiplied by the series negative 1 to the n times x to the n over 3 to the n.
01:37
And after moving 1 1 3 in 3 inside the series, we are going to get the series negative 1 to the n, x to the n, divide by 3 3...