Amit M Agarwal Solutions for Chapter: Sequence and Series, Exercise 4: Work Book Exercise 3.4

The sum of $n$ terms of the series $\frac{1}{1·2}+\frac{1}{2·3}+\frac{1}{3·4}+\dots$ is

The sum of $n$ terms of the series $\frac{3}{1^2}+\frac{5}{1^2+2^2}+\frac{7}{1^2+2^2+3^2}+\dots$ is

If $\sum _{r=1}^n{T}_r=\left(3^n-1\right),$ then find the sum of $\sum _{r=1}^n\frac{1}{T_r}$

The sum of $n$ terms of the series $5+7+13+31+85+\dots$ is

The sum of the series $\sum _{k=1}^{360}\left(\frac{1}{k\sqrt{k+1}+(k+1)\sqrt{k}}\right)$ is

The sum of the series $\sum _{r=1}^n\frac{r}{(r+1)!}$ where $n!=1·2·3\dots n,$ is

The sum of first $n$ terms of the series $1^3+3\times 2^2+3^3+3\times 4^2+5^3+3\times 6^2+\dots$ when $n$ is even, is

The sum of first $n$ terms of the series $\frac{1}{1+1^2+1^4}+\frac{2}{1+2^2+2^4}+\frac{3}{1+3^2+3^4}+\dots$ is