Question

the series $1+\frac{1}{5}+\frac{1\cdot 3}{5\cdot 10}+\frac{1\cdot 3\cdot 5}{5\cdot 10\cdot 15}+\dots$ is equal to:

• A. $\frac{1}{\sqrt{5}}$
• B. $\frac{1}{\sqrt{2}}$
• C. $\sqrt{3}$
• D. $\sqrt{\frac{5}{3}}$

Answer 1

Answer: (D)

Let $S=1+\frac{1}{5}+\frac{1\cdot 3}{5\cdot 10}+\frac{1\cdot 3\cdot 5}{5\cdot 10\cdot 15}+\dots .$
We know that
$(1+x{)}^n=1+\frac{nx}{1!}+\frac{n(n-1)}{2!}{x}^2$
$+\frac{n(n-1)(n-2)}{3!}{x}^3+\dots$
$\Rightarrow nx=\frac{1}{5}$ and $\frac{n(n-1)x^2}{2!}$
$=\frac{1\cdot 3}{5\cdot 10}$
$\Rightarrow n=-\frac{1}{2}$ and $x=-\frac{2}{5}$
$\therefore S={\left(1-\frac{2}{5}\right)}^{-1/2}$
$={\left(\frac{3}{5}\right)}^{-1/2}=\sqrt{\frac{5}{3}}$