What is the interval of convergence of #\sum_{n=0}^{oo} (\frac{1}{x(1-x)})^n#?
1 Answer
Explanation:
We can wee that
Now we know that geometric series converge when absolute value of the ratio is smaller than 1 :
So we must solve this inequality :
Let's begin with the first one :
We can easily prove that the numerator is always positive and the denominator is negetive in the interval
So this is the solution for our first inequality.
Let's see the second one :
This inequality hasas solution the interval:
So our series converge where this to intervals are both true.
Thus our interval of convergence is :