Series of Natural Numbers and other Miscellaneous Series

Let $\stackrel{r}{7\overbrace{5\dots 5}7}$ denote the $\left(r+2\right)$ digit number where the first and the last digits are $7$ and the remaining $r$ digits are $5$. Consider the sum $S=77+757+7557+\dots +\stackrel{98}{7\overbrace{5\dots 5}7}$. If $S= \frac{7{\displaystyle \stackrel{99}{\overbrace{5\dots 5 }}}7+m}{n}$, where $m$ and $n$ are natural numbers less than $3000$, then the value of $m+n$ is

Find the sum to $n$ terms of the sequence ${\left(x+\frac{1}{x}\right)}^2, {\left(x^2+\frac{1}{x^2}\right)}^2, {\left(x^3+\frac{1}{x^3}\right)}^2, ...$

Find the sum of $n$ terms of $\sum _{}\frac{1}{\left(4r^2-1\right)}$

$\underset{n\to \infty }{\lim }\sum _{r=1}^n{\cot }^{-1}\left(\frac{r^3-r+{\displaystyle \frac{1}{r}}}{2}\right)=$

Sum to $n$ terms of the series $1+2\left(1-a\right)+3\left(1-a\right)\left(1-2a\right)+4\left(1-a\right)\left(1-2a\right)\left(1-3a\right)+....+k\left(1-a\right)\left(1-2a\right)\left(1-3a\right)...\left(1-\left(k-1\right)a\right)+...$ is

The sum of the series $\sum _{r=1}^n\left(\frac{1}{\cos \theta +\cos \left(2r+1\right)\theta }\right)$ (where $n\in N)$ is equal to, $\left(\theta \ne \frac{m\mathrm{\pi }}{2}, m\in I, \left(n+1\right)\theta \ne \left(2k+1\right)\frac{\mathrm{\pi }}{2}, k\in I\right)$

The sum of the following series $1+6+\frac{9\left(1^2+2^2+3^2\right)}{7}+\frac{12\left(1^2+2^2+3^2+4^2\right)}{9}+\frac{14\left(1^2+2^2+3^2+4^2+5^2\right)}{11}+...$ upto $15$ terms is

The sum of the series ${\sec }^{-1}\left(\sqrt{2}\right)+{\sec }^{-1}\left(\frac{\sqrt{10}}{3}\right)+{\sec }^{-1}\left(\frac{\sqrt{50}}{7}\right)+....+{\sec }^{-1}\left(\sqrt{\frac{\left(n^2+1\right)\left(n^2-2n+2\right)}{{\left(n^2-n+1\right)}^2}}\right)$ is

$\sum _{n=1}^{\infty }{\tan }^{-1}\left(\frac{4n}{n^4-2n^2+2}\right)$ is equal to:

Find the sum of series ${\tan }^{-1}\left(\frac{1}{2·1^2}\right)+{\tan }^{-1}\left(\frac{1}{2·2^2}\right)+{\tan }^{-1}\left(\frac{1}{2·3^2}\right)+....+\infty$.

Find the sum to $n$ terms of the series

$\frac{3}{1^2\times 2^2}+\frac{5}{2^2\times 3^2}+\frac{7}{3^2\times 4^2}+\dots ..$

Also, find the sum of infinite terms.

$\sum _{r=1}^n{\tan }^{-1}\left(\frac{2^{r-1}}{1+2^{2r-1}}\right)$ is equal to

The value of $\sum _{k=1}^5\frac{1}{\sin \left(k+1\right)\sin \left(k+2\right)}$ is/are

Let $r^{\mathrm{th}}$ term of a series be given by $T_r=\frac{r}{1-3r^2+r^4}$ then $\underset{n\to \infty }{Lt}\sum _{r=1}^n{T}_r=$

If $\sum _{r=1}^{88}\tan r°\tan \left(r+1\right)°={\cot }^{2}1°-k$, where $k$ is a prime number, then find the difference of the digits in $k$.

Find sum of the series $\frac{n}{1·2·3}+\frac{n-1}{2·3·4}+\frac{n-2}{3·4·5}+.....$ up to n terms.

Let $a_1,a_2,a_3,\dots$ be an arithmetic progression with $a_1=7$ and common difference $8$. Let $T_1,T_2,T_3,\dots$ be such that $T_1=3$ and $T_{n+1}-T_n=a_n$ fo $n\ge 1$. Then, which of the following is/are TRUE?

If $S_n=\sum _{r=1}^n{T}_r=n\left(n+1\right)\left(n+2\right)\left(n+3\right)$, then find $\sum _{r=1}^{10}\frac{1}{T_r}$

If $\sum _{r=1}^n{t}_r=\sum _{k=1}^n\sum _{j=1}^k\sum _{i=1}^{j}2$, then $\sum _{r=1}^{10}\frac{1}{t_r}$ is equal to

If ${\cot }^{-1}\left(1^2+\frac{3}{4}\right)+{\cot }^{-1}\left(2^2+\frac{3}{4}\right)+{\cot }^{-1}\left(3^2+\frac{3}{4}\right)+\dots \dots \infty$ is equal to ${\tan }^{-1}k$, then $k$ is