Recent questions tagged summation

The value of $\sum_{i=0}^{\infty} \sum_{j=1}^{\infty} 2^{-i} 3^{-j}$ is $\_\_\_\_$. (Answer in integer)

Let $\text{C} (n, \; r) = \binom{n}{r}.$ The value of $ \displaystyle \sum_{k=0}^{20}(2k+1) \text{C} (41, 2k+1),$ is :$40 (2)^{40}$$40 (2)^{39}$$41 (2)^{40}$$41 (2)^{39}$

What is the greatest integer less than or equal to$$\sum_{n=1}^{9999}\frac{1}{\sqrt[4]{n}}?$$$1332$$1352$$1372$$1392$

How many pairs $(x,y)$ such that $x+y <= k$, where x y and k are integers and $x,y>=0, k 0$.Solve by summation rules.Solve by combinatorial argument.

The sum of the series $\dfrac{1}{1.2} + \dfrac{1}{2.3}+ \cdots + \dfrac{1}{n(n+1)} + \cdots $ is$1$$1/2$$0$non-existent

The following sum of $n+1$ terms $$2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \en...

The sum $\sum_{k=1}^n (-1)^k \:\: {}^nC_k \sum_{j=0}^k (-1)^j \: \: {}^kC_j$ is equal to $-1$$0$$1$$2^n$

Let $X=\frac{1}{1001} + \frac{1}{1002} + \frac{1}{1003} + \cdots + \frac{1}{3001}$. Then,$X \lt1$$X\gt3/2$$1\lt X\lt 3/2$none of the above holds

The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to$-1$$1$$0$does not converge

If $0 <x<1$, then the sum of the infinite series $\frac{1}{2}x^2+\frac{2}{3}x^3+\frac{3}{4}x^4+ \cdots$ is$\log \frac{1+x}{1-x}$$\frac{x}{1-x} + \log(1+x)$$\frac{1}{1-x} ...

For positive real numbers $a_1, a_2, \cdots, a_{100}$, let $$p=\sum_{i=1}^{100} a_i \text{ and } q=\sum_{1 \leq i < j \leq 100} a_ia_j.$$ Then $q=\frac{p^2}{2}$$q^2 \geq ...

Let $S=\{6, 10, 7, 13, 5, 12, 8, 11, 9\}$ and $a=\underset{x \in S}{\Sigma} (x-9)^2$ & $b = \underset{x \in S}{\Sigma} (x-10)^2$. Then$a <b$$a>b$$a=b$None of these

The smallest integer $n$ for which $1+2+2^2+2^3+2^4+ \cdots +2^n$ exceeds $9999$, given that $\log_{10} 2=0.30103$, is$12$$13$$14$None of these

Let $S=\{6,10,7,13,5,12,8,11,9\},$ and $a=\sum_{x\in S}(x-9)^{2}\:\&\: b=\sum_{x\in S}(x-10)^{2}.$ Then$a<b$$a>b$$a=b$None of these

The smallest integer $n$ for which $1+2+2^{2}+2^{3}+2^{4}+\cdots+2^{n}$ exceeds $9999$, given that $\log_{10}2=0.30103$, is$12$$13$$14$None of these

The value of $\log_{2}e-\log_{4}e+\log_{8}e-\log_{16}e+\log_{32}e-\cdots\:\:$ is$-1$$0$$1$None of these

The value of $\dfrac{1}{\log_2 n}+ \dfrac{1}{\log_3 n}+\dfrac{1}{\log_4 n}+ \dots + \dfrac{1}{\log_{2017} n}\:\:($ where $n=2017!)$ is$1$$2$$2017$none of these

The value of $\dfrac{x}{1-x^2} + \dfrac{x^2}{1-x^4} + \dfrac{x^4}{1-x^8} + \dfrac{x^8}{1-x^{16}}$ is$\frac{1}{1-x^{16}}$$\frac{1}{1-x^{12}}$$\frac{1}{1-x} – \frac{1}{1-x^...

$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}$ is$2$$1$$\infty$not a convergent series

Let $A=\begin{pmatrix} -1 & 2 \\ 0 & -1 \end{pmatrix}$, and $B=A+A^2+A^3+ \dots +A^{50}$. Then$B^2 =1$$B^2 =0$$B^2 =A$$B^2 =B$