Example: Find out the radius of convergence for the following power series.

Solution:

Therefore, in this case we have,

a_n = ((-3)ⁿ)/(n7ⁿ⁺¹)  

a_n+1 = ((-3)ⁿ⁺¹)/(n7ⁿ⁺²)  

Keep in mind that to calculate aⁿ⁺¹ all we do is replace all the n's in a_n along with n+1. By using the ratio test then provides,

= (3/7) |x -5|

Here we know that the series will converge if,

(3/7) |x -5| < 1                                    ⇒                     |x -5| < 7/3

As well as the series will diverge if,

3/7) |x -5| > 1                         ⇒                     |x -5| > 7/3

Conversely, the radius of the convergence for the series is,

r = 7/3