Recent questions tagged summation

+3 votes

2 answers

1

ISI2014-DCG-4
$\underset{n \to \infty}{\lim} \dfrac{1}{n} \bigg( \dfrac{n}{n+1} + \dfrac{n}{n+2} + \cdots + \dfrac{n}{2n} \bigg)$ is equal to $\infty$ $0$ $\log_e 2$ $1$

asked Sep 23 in Calculus by Arjun Veteran (424k points) | 92 views

+1 vote

2 answers

2

ISI2014-DCG-16
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

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 45 views

+1 vote

1 answer

3

ISI2014-DCG-23
The sum of the series $\:3+11+\dots +(8n-5)\:$ is $4n^2-n$ $8n^2+3n$ $4n^2+4n-5$ $4n^2+2$

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 30 views

+1 vote

1 answer

4

ISI2014-DCG-34
The following sum of $n+1$ terms $2 + 3 \times \begin{pmatrix} n \\ 1 \end{pmatrix} + 5 \times \begin{pmatrix} n \\ 2 \end{pmatrix} + 9 \times \begin{pmatrix} n \\ 3 \end{pmatrix} + 17 \times \begin{pmatrix} n \\ 4 \end{pmatrix} + \cdots$ up to $n+1$ terms is equal to $3^{n+1}+2^{n+1}$ $3^n \times 2^n$ $3^n + 2^n$ $2 \times 3^n$

asked Sep 23 in Combinatory by Arjun Veteran (424k points) | 33 views

+1 vote

0 answers

5

ISI2014-DCG-65
The sum $\dfrac{n}{n^2}+\dfrac{n}{n^2+1^2}+\dfrac{n}{n^2+2^2}+ \cdots + \dfrac{n}{n^2+(n-1)^2} + \cdots \cdots$ is $\frac{\pi}{4}$ $\frac{\pi}{8}$ $\frac{\pi}{6}$ $2 \pi$

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 39 views

0 votes

1 answer

6

ISI2014-DCG-72
The sum $\sum_{k=1}^n (-1)^k \:\: {}^nC_k \sum_{j=0}^k (-1)^j \: \: {}^kC_j$ is equal to $-1$ $0$ $1$ $2^n$

asked Sep 23 in Combinatory by Arjun Veteran (424k points) | 28 views

+2 votes

2 answers

7

ISI2015-MMA-10
The value of the infinite product $P=\frac{7}{9} \times \frac{26}{28} \times \frac{63}{65} \times \cdots \times \frac{n^3-1}{n^3+1} \times \cdots \text{ is }$ $1$ $2/3$ $7/3$ none of the above

asked Sep 23 in Calculus by Arjun Veteran (424k points) | 35 views

+1 vote

1 answer

8

ISI2015-MMA-17
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

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 16 views

0 votes

1 answer

9

ISI2015-MMA-24
The series $\sum_{k=2}^{\infty} \frac{1}{k(k-1)}$ converges to $-1$ $1$ $0$ does not converge

asked Sep 23 in Numerical Ability by Arjun Veteran (424k points) | 14 views

+1 vote

1 answer

10

ISI2015-MMA-54
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 $\log \frac{1+x}{1-x}$ $\frac{x}{1-x} + \log(1+x)$ $\frac{1}{1-x} + \log(1-x)$ $\frac{x}{1-x} + \log(1-x)$

asked Sep 23 in Others by Arjun Veteran (424k points) | 11 views

0 votes

0 answers

11

ISI2015-MMA-80
Let $0 < \alpha < \beta < 1$. Then $ \Sigma_{k=1}^{\infty} \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x} dx$ is equal to $\log_e \frac{\beta}{\alpha}$ $\log_e \frac{1+ \beta}{1 + \alpha}$ $\log_e \frac{1+\alpha }{1+ \beta}$ $\infty$

asked Sep 23 in Calculus by Arjun Veteran (424k points) | 14 views

+1 vote

0 answers

12

ISI2015-MMA-84
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 $q=\frac{p^2}{2}$ $q^2 \geq \frac{p^2}{2}$ $q< \frac{p^2}{2}$ none of the above

asked Sep 23 in Others by Arjun Veteran (424k points) | 10 views

0 votes

1 answer

13

ISI2015-DCG-2
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

asked Sep 18 in Verbal Ability by gatecse Boss (16.8k points) | 35 views

0 votes

1 answer

14

ISI2015-DCG-15
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

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 20 views

0 votes

2 answers

15

ISI2016-DCG-2
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

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 38 views

0 votes

1 answer

16

ISI2016-DCG-17
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

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 9 views

0 votes

1 answer

17

ISI2016-DCG-23
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

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 12 views

0 votes

1 answer

18

ISI2016-DCG-65
The value of $\sin^{2}5^{\circ}+\sin^{2}10^{\circ}+\sin^{2}15^{\circ}+\cdots+\sin^{2}90^{\circ}$ is $8$ $9$ $9.5$ None of these

asked Sep 18 in Geometry by gatecse Boss (16.8k points) | 10 views

0 votes

1 answer

19

ISI2017-DCG-1
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

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 17 views

0 votes

0 answers

20

ISI2017-DCG-13
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 $\frac{1}{1-x^{16}}$ $\frac{1}{1-x^{12}}$ $\frac{1}{1-x} – \frac{1}{1-x^{16}}$ $\frac{1}{1-x} – \frac{1}{1-x^{12}}$

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 10 views

0 votes

1 answer

21

ISI2018-DCG-27
$\sum_{n=1}^{\infty}\frac{1}{n(n+1)}$ is $2$ $1$ $\infty$ not a convergent series

asked Sep 18 in Numerical Ability by gatecse Boss (16.8k points) | 6 views

+2 votes

1 answer

22

Kenneth Rosen Edition 6th Exercise 2.4 Question 15c (Page No. 161)
$\sum_{j=2}^{8}(-3)^j$

asked Dec 5, 2018 in Set Theory & Algebra by aditi19 Active (5.1k points) | 71 views

+2 votes

1 answer

23

ISI2016-MMA-18
Let $A=\begin{pmatrix} -1 & 2 \\ 0 & -1 \end{pmatrix}$, and $B=A+A^2+A^3+ \dots +A^{50}$. Then $B^2 =1$ $B^2 =0$ $B^2 =A$ $B^2 =B$

asked Sep 13, 2018 in Linear Algebra by jothee Veteran (105k points) | 18 views

0 votes

0 answers

24

ISI2016-MMA-22
The infinite series $\Sigma_{n=1}^{\infty} \frac{a^n \log n}{n^2}$ converges if and only if $a \in [-1, 1)$ $a \in (-1, 1]$ $a \in [-1, 1]$ $a \in (-\infty, \infty)$

asked Sep 13, 2018 in Others by jothee Veteran (105k points) | 10 views

+1 vote

1 answer

25

Infinite series
Find the infinite sum of the series $1 + \frac{4}{7} + \frac{9}{7^2} + \frac{16}{7^3} + \frac{25}{7^4} + .............\Join$

asked Aug 8, 2018 in Numerical Ability by pankaj_vir Boss (10.6k points) | 153 views

0 votes

0 answers

26

Bounding Summation
How does the below bounds to logn? Please explain the steps 1 and 2. I came to know that they are using the idea of splitting the summations and bounding them. How the first and second step came?

asked May 11, 2018 in Algorithms by Ayush Upadhyaya Boss (27.6k points) | 138 views

+1 vote

1 answer

27

addition
value of 1/3 + 1/15 + 1/35 +............................+1/9999 a)100/101 b)50/101 c)100/51 d)50/51

asked Sep 12, 2017 in Numerical Ability by A_i_$_h Boss (10.1k points) | 437 views

+5 votes

1 answer

28

Series Summation
Series summation of $S_n$ in closed form? $\begin{align*} &S_n = \frac{1}{1.2.3.4} + \frac{1}{2.3.4.5} + \frac{1}{3.4.5.6} + \dots + \frac{1}{n.(n+1).(n+2).(n+3)} \end{align*}$

asked Jun 11, 2017 in Set Theory & Algebra by dd Veteran (57k points) | 219 views

+3 votes

2 answers

29

Manipulation of sum
Prove the identity: $\begin{align*} &\sum_{i=0}^{n}\sum_{j=0}^{i} a_ia_j = \frac{1}{2}\left ( \left ( \sum_{i=0}^{n}a_i \right )^2 + \left ( \sum_{i=0}^{n}a_i^2 \right )\right ) \end{align*}$

asked Feb 25, 2017 in Combinatory by dd Veteran (57k points) | 169 views

+2 votes

0 answers

30

summation series
what is the summation of this series? S=nC0*20+nC1*21+nC2*22+..............nCn*2n

asked Jan 17, 2017 in Combinatory by firki lama Junior (681 points) | 101 views

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