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
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