**Convergence Of infinite Series**

Here

U_n=\frac{1.2.3….n}{3.5.7….(2n+1)} \, \; and \, \;U_{n+1}=\frac{1.2.3….n(n+1)}{3.5.7….(2n+1)(2n+3)}

\frac{U_n}{U_{n+1}}=\frac{\frac{1.2.3….n}{3.5.7….(2n+1)}}{\frac{1.2.3….n(n+1)}{3.5.7….(2n+1)(2n+3)}}

=\frac{2n+3}{n+1}

Therefore

\lim_{n\rightarrow \infty}\frac{U_n}{U{n+1}}=\lim_{n\rightarrow \infty}\frac{2n+3}{n+1}\ \ \ \lim_{n\rightarrow \infty}\frac{2+\frac{3}{n}}{1+\frac{1}{n}}=2> 1

**Hence By Ratio Test The given Series is Convergent**

keywords

Infinite Series Convergence

Convergence/Divergence of Series

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\sum r^{n-1}=1+r+r^2+r^3+r^4+...

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