00:01
In this question, we are asked to find the series for x over 10x squared plus 1, and then find the interval of convergence.
00:08
To find the series for f, we will start with a series for 1 over 1 minus x, which equals to the series, x to the n, n from 0 to infinity.
00:20
Now, we can rewrite the function f as x multiplied by 1 over 1 minus negative 10 x squared.
00:32
Now the fraction 1 over 1 minus negative 10 x squared is similar to 1 over 1 minus x with x replaced by negative 10 x squared.
00:46
This means we can rewrite the series for 1 over 1 minus negative 10 x squared.
00:54
We can get the series for this function by replacing x by negative 10 x squared in the previous series.
01:08
And we're going to get the series negative 1 to the n, 10 to the n, x to the 2n.
01:20
After plugging in this series in f, we're going to get x times the series negative 1 to the n, 10 to the n, 10 to the n, x to the 2n.
01:35
And this equals to the series, negative 1 to the n, 10 to the n, x to the 2n plus 1...