Question

Find the value of the series
$2+\frac{5}{|\underset{–}{2}.3}+\frac{5.7}{|\underset{–}{3}.3^2}+\frac{\mathrm{5.7.9}}{|\underset{–}{4}.3^3}+...$

Answer 1

given the series

$y=2+\frac{5}{2!.3}+\frac{5.7}{3!{.3}^2}+\frac{\mathrm{5.7.9}}{4!3^3}+.......$

above series can be manipulated as

$y=1+\frac{3}{2}.\frac{2}{3}+\frac{\frac{3}{2}.\frac{5}{2}}{2!}.\frac{2^2}{3^2}+\frac{\frac{3}{2}.\frac{5}{2}.\frac{7}{2}{.2}^3}{3!{.3}^3}+\frac{\frac{3}{2}.\frac{5}{2}.\frac{7}{2}.\frac{9}{2}}{4!}.\frac{2^4}{3^4}+.......$

clearly above series is of the form

$(1+x{)}^n$ where n is negative or fraction

$x=\frac{-2}{3}$ & $n=\frac{-3}{2}$

$y=(1-\frac{2}{3}{)}^{\frac{-3}{2}}$

$y=(\frac{1}{3}{)}^{\frac{-3}{2}}$

$y=(3{)}^{\frac{3}{2}}=3\sqrt{3}$