Determine if the following series is convergent or divergent by using Radio Test.

Solution

With this first illustration let's be a little careful and ensure that we have everything down correctly.  Now Here are the series terms a_n.

a_n = (-10)ⁿ/4²ⁿ⁺¹ (n+1)

Remind that to compute a_n+1 all that there is a requirement to do is substitute n+1 for all the n's in a_n.

a_n+1 = (-10)ⁿ⁺¹/ 4^2(n+1)+1 ((n+1)+1

= (-10)ⁿ⁺¹/ 4²ⁿ⁺³ (n+2)

Now, to illustrate L we will use,

As this will be a little easier while dealing with fractions like we've got here.  Thus,

Thus, L<1 and so by the Ratio Test the series converges absolutely and therefore will converge.