Test The Following Series 1/3+1.2/3.5+1.2.3/3.5.7+…∞

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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

Show that the Geometric series

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

where r > 0 , is convergent if r < 1and diverges if r ≥ 1| Geometric series solved problems

If \sum a_n is divergent and \sum b_n is convergent, show that \sum (a_n-b_n) is divergent


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