Miscellaneous Series

Mention any two properties of $\mathrm{Pi}$ Notation.

Find the sum, $6+8+10+\cdots +72$.

Find the sum, $20+22+24+\cdots +72$.

Find the sum, $20+22+24+\cdots +96$.

Find the sum, $18+20+22+24+\cdots +102$.

Find the sum, $18+20+22+24+\cdots +60$.

The value of $\cot \left[\sum _{n=1}^{19}{\cot }^{-1}\left(1+\sum _{p=1}^{n}2p\right)\right]$ is

Let $m$ and $k$ be positive integers $(m\ne k)$, then $\underset{n\to \infty }{\lim }n\left\{{\left(1+\frac{1}{n}\right)}^m+{\left(1+\frac{2}{n}\right)}^m+\dots +{\left(1+\frac{k}{n}\right)}^m-k\right\}$ is

If the sum of the first $15$ terms of the series $3+7+14+24+37+\dots$ is $15k$, then $k$ is equal to

If $\sum _{k=1}^n\varphi \left(k\right)=\frac{2n}{n+1}$, then $\sum _{k=1}^{10}\frac{1}{\varphi \left(k\right)}$ is equal to

Let $S_n$ be the sum of the first $n$ terms of the series $1^2+2·5^2+3^2+2·7^2+5^2+2·9^2+\dots \dots ..$ If $S_{20}=20A$, then $A$ is equal to

Sum of the series $1\left(1\right)+2\left(1+3\right)+3\left(1+3+5\right)+4\left(1+3+5+7\right)+\dots +10\left(1+3+5+7+\dots +19\right)$ is equal to

The sum of odd integers from $1$ to $2001$ is

What is the value of  $2^2+4^2+6^2+\cdots +{20}^2?$

The value of  $1.2.3+2.3.4+3.4.5+\dots n$ terms is equal to

Let $S_n$ denote the sum of the cubes of the first $n$ natural numbers and $s_n$ denote the sum of the first $n$ natural numbers. Then $\sum _{r=1}^n\frac{S_r}{s_r}$ equals

Let $S_n=\sum _{k=0}^n\left(\frac{1}{\sqrt{k+1}+\sqrt{k}}\right)$. What is the value of $\sum _{n=1}^{99}\left(\frac{1}{S_n+S_{n-1}}\right) ?$

Let $x_1, x_2, \dots , x_{2014}$ be real numbers different from $1,$ such that $x_1+x_2+\cdots +x_{2014}=1$ and $\frac{x_1}{1-x_1}+\frac{x_2}{1-x_2}+\cdots +\frac{x_{2014}}{1-x_{2014}}=1$. What is the value of $\frac{x_1^2}{1-x_1}+\frac{x_2^2}{1-x_2}+\frac{x_3^2}{1-x_3}+\cdots +\frac{x_{2014}^2}{1-x_{204}} ?$

If the sum of the series $\frac{1^3}{1}+\frac{1^3+2^3}{1+3}+\frac{1^3+2^3+3^3}{1+3+5}+.......$ upto $16$ terms is $S_{16},$ then find $\left[\frac{S_{16}}{10}\right]$, where $[·]$ represents greatest integer function.

Consider the arithmetic series $S=1+3+5+7+\dots +61$

Write the series using sigma notation.