How do you find the interval of convergence [Math Processing Error] from [Math Processing Error]?

1 Answer
Mar 7, 2017

[Math Processing Error]

Explanation:

The series [Math Processing Error] converges when [Math Processing Error]. We can find [Math Processing Error] then find the values of [Math Processing Error] that satisfy [Math Processing Error].

[Math Processing Error]

We can simplify and move the [Math Processing Error] terms out of the limit, since the limit depends only on how [Math Processing Error] changes.

[Math Processing Error]

The limit approaches [Math Processing Error]:

[Math Processing Error]

The series converges when [Math Processing Error]:

[Math Processing Error]

[Math Processing Error]

Before we say we're done, we have to check the endpoints [Math Processing Error] and [Math Processing Error] by plugging them into the original series and seeing if the series does in fact converge. If the series converges for the endpoint, the endpoint will be included in the interval.

At [Math Processing Error], the series in question is:

[Math Processing Error]

This is divergent through the p-series test, since [Math Processing Error]. Thus [Math Processing Error] is not included in the interval of convergence.

At [Math Processing Error], we have

[Math Processing Error]

Through the alternating series test, we see that this does converge, so [Math Processing Error] will be included in the interval of convergence.

The interval is then:

[Math Processing Error]