Find out the radius of convergence, Mathematics

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

1580_Find out the radius of convergence.png

Solution:

Therefore, in this case we have,

an = ((-3)n)/(n7n+1)  

an+1 = ((-3)n+1)/(n7n+2)  

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

2359_Find out the radius of convergence1.png

= (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


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