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`sum_(n=1)^oo (4x)^n/n^2` Find the radius of convergence of the power series.

`sum_(n=1)^oo (4x)^n/n^2`


To find radius of convergence of a series `sum` `a_n` , apply the Ratio Test. 


`L = lim_(n->oo) |a_(n+1)/a_n|`



`L=lim_(n->oo) | (4x)^(n+1)/(n+1)^2 * n^2/(4x)^n|`


`L= lim_(n->oo) |(4xn^2)/(n+1)^2|`


`L = |4x| lim_(n->oo) |n^2/(n+1)^2|`


`L = |4x| * 1`


`L = |4x|`


`L =4|x|`


Take note that in Ratio Test, the series converges when L < 1. 


`L < 1`


`4|x| lt 1`


`|x|lt1/4`


Therefore, the radius of convergence of the given series is `R = 1/4` .

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