Find the values of #x# for which the following series is convergent?
When trying to determine the radius and/or interval of convergence of power series such as these, it is best to use the Ratio Test, which tells us for a series
For Power Series, however, three cases are possible
a. The power series converges for all real numbers; its interval of convergence is
b. The power series converges for some number
c. The most frequent case, the power series converges for
So, apply the Ratio Test:
Now, let's determine the interval:
We need to plug
Therefore, the series converges for
We can use the ratio test which says that if we have a series
it is definitely convergent if:
In our case,
So, we need to check when
I made a mistake here, but the above answer has the same method and a correct answer, so just have a look at that instead.