The coefficients of the power series \(\displaystyle \sum_{n=0}^{\infty} a_n(x-2)^n \) satisfy \(a_0=5\) and \(a_n=\left(\dfrac{2n+1}{3n-1}\right)a_{n-1} \) for all \(n\geq 1\). The radius of convergence of the series is

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We are given the recursive formula:

$$ a_n=\left(\frac{2n+1}{3n-1} \right)a_{n-1} $$

This implies a geometric sequence with ratio \(\dfrac{2n+1}{3n-1}\). This ratio approaches \(\dfrac{2}{3}\) as \(n\) approaches infinity.

Therefore,

$$ \left| \frac{2}{3}(x-2) \right| \lt 1 $$ $$ -\frac{3}{2} \lt x-2 \lt \frac{3}{2} $$