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)

Therefore, we get the radius of convergence as 3 and the centre as .

Using (2), we can write the interval of convergence as .

Step 3 of 5</p>

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

At , we have the series as

.

Now, using the nth term test for divergence, we can see that .

Therefore, the series diverges.

At , we have the series as

.

Now, using the nth term test for divergence, we can see that .

Therefore, the series diverges.

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