Intervals of Convergence

In Exercise , (a) find the series’ radius and interval of convergence. For what values of x does the series converge (b) absolutely, (c) conditionally?

Step 1 of 5</p>

Here, we have to find

The series’ radius and interval of convergence.The value of x for which the series converges absolutely.The value of x for which the series converges conditionally..

Step 2 of 5</p>Let us use the ratio test in order to find the radius of convergence.

Therefore, according to the ratio test, a series is said to be absolutely convergent if

…..(1)

Therefore, according to our question, we have .

Thus,

Thus, using (1), we have …..(2)

On dividing the equation (2) by 4, we get

.

Thus, we get the radius of convergence as and the centre .

Now, we need to find the interval of convergence.

Consider the equation (2), .

..(3)

Using (3), we have found that the interval of convergence is .

Step 3 of 5</p>

Now let us test the convergence of the series at the endpoints.

At , we have the series as

.

We can see that the series is absolutely convergent using the p-series test..

Therefore, the series converges at .

At , we have the series as

.

Now, using the p series test, we can see that the given series is convergent at .

Thus, we can see that the series does not converge conditionally for any value of x.