Questions about evaluating summations, especially finite summations. For infinite series, please consider the (sequences-and-series) tag instead.

learn more… | top users | synonyms

0
votes
0answers
34 views

Need to understand this summation with max notation

Firstly, apologies needed for my math description if it does not sound right. I have come across a paper where I saw a summation notation with a max function in it which I am little confused to ...
3
votes
2answers
49 views

Another Hockey Stick Identity

I know this question has been asked before and has been answered here and here. I have a slightly different formulation of the Hockey Stick Identity and would like some help with a combinatorial ...
4
votes
3answers
104 views

Find $\frac{1}{7}+\frac{1\cdot3}{7\cdot9}+\frac{1\cdot3\cdot5}{7\cdot9\cdot11}+\cdots$ upto 20 terms

Find $S=\frac{1}{7}+\frac{1\cdot3}{7\cdot9}+\frac{1\cdot3\cdot5}{7\cdot9\cdot11}+\cdots$ upto 20 terms I first multiplied and divided $S$ with $1\cdot3\cdot5$ $$\frac{S}{15}=\frac{1}{1\cdot3\cdot5\...
3
votes
3answers
46 views

Explanation of the Sum of an Infinite Series Equation

I've been presented with the following infinite sum (where $P$ is the probability of an event, and $1-P$ is therefore the probability of it not occurring. I was given the following equation as fact: $...
1
vote
1answer
41 views

The analytic extension of $\sum_{k=1}^n\frac1k$ and an induction

The analytic extension of the sum of the first $n$ reciprocals is given as $$\sum_{k=1}^n\frac1k=\int_0^1\frac{x^n-1}{x-1}dx$$ I am wondering if $\frac1{n+1}+\sum_{k=1}^n\frac1k=\sum_{k=1}^{n+1}\...
1
vote
0answers
25 views

Limit of a certain sum

I need to show that $$\sum_{i=0}^{m} \binom{m}{m-i}\binom{m^2-m}{i} (1-p)^{\binom{i}{2} + i m} \bigg/ \binom{m^2}{m} (1-p)^{\binom{m}{2}} \to 0$$ as $m \to \infty$, where $p = \frac{1}{m}$, and the ...
0
votes
0answers
35 views

Calculating infinite series for a hospital waiting queue

For my project, I had to simulate a hospital waiting queue, and ended up stuck with this equation. $$ 1=\sum_{i=0}^\infty \left(\frac{\lambda}{\mu+i\gamma}\right)^iP_0 $$ Could any kind soul help ...
0
votes
0answers
63 views

Prove with induction that $\sum_{k=0}^{n-1}x^{k}=\frac{x^n-1}{x-1}$

Prove with induction that $$\sum_{k=0}^{n-1}x^{k}=\frac{x^n-1}{x-1}$$ It seems simple but I have tried for I don't know how long by now... Anyone can manage this?
0
votes
0answers
10 views

Can you use Dirichlet's hyperbola method with any of these pathological logarithms?

I would like to learn Dirichlet's hyperbola method in some of myself next posts. I know its meaning and relationship with the divisor function and lattice problems, but in this ocassion I want to ...
3
votes
5answers
66 views

How to prove that $\sum_{i=j}^nn-i = \sum_{i=1}^{n-j}i$?

Trying to solve question 2-3 from Skiena's Algorithm Design Manual which asks to find the runtime of the following loop: ...
13
votes
6answers
211 views

How to prove that $\sum_{i=0}^n 2^i\binom{2n-i}{n} = 4^n$.

So I've been struggling with this sum for some time and I just can't figure it out. I tried proving by induction that if the sum above is a $S_n$ then $S_{n+1} = 4S_n$, but I didn't really succeed so ...
1
vote
3answers
734 views

How to prove the sum of combination is equal to $2^n - 1$

One of the algorithm I learnt involve these steps: $1$. define a set $S$ of $n$ elements $2$. form a subset $S'$ of $k$ choice from $n$ elements of the set $S$ ($k$ starts with $1$), which is ...
0
votes
2answers
58 views

Looking for a closed form for $a_m =\sum_{k=0}^{\infty}\binom{k+m}{m}\frac{1}{4^{k}(2(k+m))!}$

I have this sequence $$ a_m =\sum_{k=0}^{\infty}\binom{k+m}{m}\frac{1}{4^{k}(2(k+m))!} $$ and there seems to exist a patern arising when it is evaluated by WA. It involves $\cosh(1/2)$ and $\sinh(1/2)...
0
votes
2answers
35 views

Value of n summations of 1 $\sum_{0 \le a_1\lt a_2 \lt a_3 …\lt a_k \le n}1$

I need to find $$\sum_{0 \le a_1\lt a_2 \lt a_3 ...\lt a_k \le n}1$$ My attempt: I think it is equal to $ ^nC_k $ as $$\sum_1^n{1} = n = ^nC_1$$ $${\sum \sum }_{0\le i \lt j \le n} 1 = \frac {n(n-1)}...
1
vote
3answers
34 views

Evaluate $\int_1^N \frac{-3N+6t-3}{t^3(N-t+1)^4}dt$ when $N=3$ or $N=5$

Let the Cauchy product $$(\zeta(3))^2=\sum_{n=1}^\infty c_n,$$ where $$c_n=\sum_{k=1}^n\frac{1}{k^3(n-k+1)^3},$$ and $\zeta(3)$ is the Apèry constant. Taking $f(x)=\frac{1}{x^3(N-x+1)^3}$ in Abel's ...
1
vote
3answers
69 views

Matlab sum is wrong: Double symsum gives incorrect result

I'm trying to calculate the double sum $$ \frac{1}{10} \sum_{x=1}^{10} \left( \frac{1}{x} \sum_{n=0}^{floor(log_{10}x)} 10^n \right).$$ In MATLAB, my result is ...
1
vote
1answer
21 views

Finding $\sum_{j=m}^{n}\frac{a^{j-m}}{N-j}$

How can we tackle $$\sum_{j=m}^{n}\frac{a^{j-m}}{N-j}$$ when $0<m<n<N$. I have been using Euler-Maclaurin sum to change this object to some integral but it gets a bit messy. I appreciate any ...
1
vote
0answers
29 views

Series with Markov Chains Probabilities

Notation For each $t \in \mathbb{N}$, let $h_t \in H$ be a random variable that follows a Markov chain, and $h^t \equiv \{h_0,h_1,\dots,h_t\} \in H^t$. Let $\Pi(h^{t})$ be the probability that a ...
0
votes
1answer
28 views

How can you simplify this expression for amount of triangles?

I was given a question as a challenge were I was supposed to find a formula to find how many triangles there are when you draw $n$ and $m$ amount of lines from points $N$ and $M$ to the opposite sides....
1
vote
1answer
28 views

Do not understand algebra technique used to computer summation

I am going through a practice exam for my Discrete Mathematics class and do not understand the algebra used in the following summation computation. Summation to compute: Answer: What I don't ...
0
votes
1answer
25 views

Limit of a series with a lot of dependencies

Let $n \rightarrow \infty$ and consider $$\sum_{x=\lfloor \log^6(n)\rfloor}^{\lceil \frac{n}{\log^2(n)}\rceil} \left(\frac{n}{\log^2(n) x} e^{-\frac{\log^{16}(n)}{n}}\right)^x$$ Do we know anything ...
0
votes
2answers
27 views

Help finishing proof for $\sum_{i=1}^{n-1} (-1)^{i+1}i! \leq \frac{(2n)!}{2}$

I need help finishing this proof. I've come to a point where I don't know how to continue. I need to prove that the following inequality is true for all positive integers $n$. $$\sum_{i=1}^{n-1} (-1)^...
4
votes
0answers
56 views

Calculating $\lim_{n \to \infty}\left(\dfrac{1^n +2^n +3^n + \ldots + n^n}{n^n}\right)$ [duplicate]

Evaluate : $$\lim_{n \to \infty}\left(\dfrac{1^n +2^n +3^n + \ldots + n^n}{n^n}\right)$$ I tried using squeeze theorem, but I couldn't find the proper inequality. I also thought of using ...
0
votes
0answers
25 views

Compute limit of a double-sum

Let $n \rightarrow \infty$. I would like to compute the limit $$\sum_{x,y=1/2 \log^c(n)}^{\frac{n}{log^{d}(n)}} e^{(x+y)\log n}e^{-C x y \log^c(n)}$$ where $c,d,C \in \mathbb{N}$ can be chosen ...
0
votes
3answers
59 views

Request for a proof that $\sum\limits_{i=1}^n i^{k+1}=(n+1)\sum\limits_{i=1}^n i^k-\sum\limits_{p=1}^n\sum\limits_{i=1}^p i^k$

Prove $$\sum_{i=1}^n i^{k+1}=(n+1)\sum_{i=1}^n i^k-\sum_{p=1}^n\sum_{i=1}^p i^k \tag1$$ for every integer $k\ge0$. By principle of induction, $$\sum_{i=1}^n i = n(n+1)- \sum_{p=1}^n p$$ $$2\...
0
votes
1answer
17 views

double summation notation

In a paper I am studying, the author writes $$\sum_{{i=1}\atop {k=1}}^{N+1} C_i \eta_k$$ How are the two indices to be interpreted? In other words, how would this expression be written using sigma ...
0
votes
1answer
66 views

How to prove this quasi-geometric trigonometric series identity without induction

$$\frac{2}{\sin{x}}\sum_{r=1}^{n-1} \sin{rx}\cos{[(n-r)y]} \equiv \frac{\cos{(nx)}-\cos{(ny)}}{\cos{x}-\cos{y}} - \frac{\sin{(nx)}}{\sin{x}}$$ The identity can be tediously proven using the Axiom of ...
1
vote
1answer
20 views

On computations around $\sum_{n=1}^N\frac{n\Lambda(n)}{n+N}$, where $\Lambda(n)$ is von Mangoldt function

By specialization with $F(x)=\frac{1}{1+x}$ in Apostol's Theorem 4.17 (Apostol, Introduction to Analytic Number Theory (Springer)), for intergers $N\geq 1$ one has $$\frac{\log N}{1+N}+\sum_{n=1}^N\...
1
vote
1answer
34 views

Sigma notation for following nested loop

How may the following programming statement be written as summation? ...
1
vote
1answer
43 views

Extending binomial identity $ \sum\limits_{k=0}^n\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$ to $0<x<1$

I found in Matlab that $$ \sum_{k=0}^n~\frac{(-1)^k}{k+x}\binom{n}{k}\binom{n+k}{k}=0$$ for $1\leq x< n$ only (I am about 95% sure of this since the sum is numerically unstable and cannot give ...
1
vote
0answers
24 views

Calculation of renewal function $R(t) = \sum{F^n(t)}$?

My textbook defines the renewal function $R(t) = E[N_t] = \sum_{n=0}^\infty F^n(t)$, where $F^n(t)$ appears to be the n-fold convolution of $F$ with itself. $F$ is the distribution of the interrenewal ...
1
vote
1answer
20 views

Pull a term out of a double sum

I'm trying to rewrite a double sum in the following format $$ \sum_{l=0}^\infty \sum_{n=0}^\infty z^{n-2l} g(n,l) = \sum_{m=-\infty}^\infty z^m h(n) $$ For some $h(n)$, which will probably involve a ...
0
votes
3answers
40 views

Explanation of sums

How would you explain this notation in plain English? I am having issues getting my head around the meaning of the summation symbol. $$ x = \frac1n\sum_{i=1}^n x_i\;. $$
0
votes
1answer
29 views

Can a sum over all the partitions be reduced to a non-partition sum?

Let's say we have $$\sum_{k=1}^{p(N)}\prod_{j=1}^lf(i_{j_k},r_{j_k})$$ for some two-variable function $f(x,y)$. Let $\lambda_k$ be the $k\text{th}$ partition of the integer $N$ into $l_k$ distinct ...
0
votes
2answers
46 views

Calculate the partial sum of an series

Good morning, i have a problem when i go to calculate the partial sum of this series: $S = 2+\frac{2}{3}+\frac{2}{9}+\frac{2}{27}+...+\frac{2}{3^{n-1}}$ I make this: If this an geometric series ...
1
vote
0answers
25 views

Understanding a theorem about double series

In baby Rudin, there is a theorem stated that: Given a double series $\{a_{ij}\},i=1,2,3,...,j=1,2,3,...,$ suppose that $$\sum_{j=1}^{\infty}|a_{ij}|=b_{i}$$ and $\sum b_i$ converges. Then $$\...
0
votes
3answers
37 views

Sum of all distinct numbers made

Question: Find the sum of all distinct four digit numbers that can be formed using the digits 1; 2; 3; 4; and 5, each digit appearing at most once. I have no clue as to where to begin this question. ...
2
votes
1answer
51 views

For which values of $p$ the series $\sum_{n = 2}^{\infty}\frac{1}{\ln^p{n}}$ converges?

I'm trying to find all values of $p$ for which the following series converges: $$\sum_{n = 2}^{\infty}\frac{1}{\ln^p{n}}$$ So my first approach was to use the integral test because $\frac{1}{\ln^p{n}...
0
votes
0answers
13 views

Describe the characteristics of this function (where it is defined, continuous, diff, twice diff).

Consider the function $f(x)=\sum\limits_{k=1}^{\infty}\frac{\sin(x/k)}{k}$ Where is $f$ defined? Continuous? Differentiable? Twice-differentiable? My thoughts so far: Initially I thought that due ...
1
vote
1answer
65 views

Easy geometric sum with binomial coefficient

In the context of stochastic processes I came across the following equality, where $|s| < 1, p \in [0,1]$: $$\sum^\infty_{k=0}(s^2p(1-p))^k\begin{pmatrix} 2k \\ k \end{pmatrix} = \frac{1}{\sqrt{1-...
1
vote
0answers
33 views

Under what conditions can I split a power of a binomial sum into two products?

I was reading a paper and came across a section that claimed that if $y \in \mathbb{N}$, and if $x \in [0,1]$, then for the expression: $$ \left(\frac{1}{2}+ \frac{x}{4}\right)^y $$ there exists a $...
3
votes
2answers
69 views

Proof of $1^3+1^3+2^3+3^3+5^3+\cdots +F_n^3=\frac{F_nF_{n+1}^2+(-1)^{n+1}[F_{n-1}+(-1)^{n+1}]}{2}$

Fibonacci series $F_0=0$, $F_1=1$; $F_{n+1}=F_n+F_{n-1}$ This is a well known identity $1^2+1^2+2^2+3^2+5^2+\cdots +F_n^2=F_nF_{n+1}$ I was curious and look every websites for a closed form of $1^...
1
vote
1answer
64 views

Mathematical induction proof problem: $\sum_{i=1}^{n-1} i(i+1) = \frac{n(n+1)(n-1)}3$

I am having difficulty proving the inductive hypothesis $(k+1)$ for the following statement: $$\sum_{i=1}^{n-1} (i(i+1)) = \frac{(n)(n+1)(n-1)}{3}$$ This is what I have so far: $$(Step \ 1) \sum_{i=...
1
vote
0answers
24 views

How to isolate and solve for k in a Sigma notation probability mass function equation?

"isolate and solve for k:" $$P(X = k) = \sum_{k=0}^n {{{K \choose k} {{N-K} \choose {n-k}}}\over {N \choose n}}$$ If the above equation is a function of P, how would the equation be stated as a ...
4
votes
2answers
83 views

Prove $\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$

I want to prove $$\prod_{k=1}^n(1+a_k)\leq1+2\sum_{k=1}^n a_k$$ if $\sum_{k=1}^n a_k\leq1$ and $a_k\in[0,+\infty)$ I have no idea where to start, any advice would be greatly appreciated!
0
votes
1answer
25 views

On a bound about $\sum_{n\leq n}\sqrt{\frac{x}{n}} \left[\sqrt{\frac{n}{x}} M \left(\frac{x}{n} \right) \right] $

From the fact that $f(x)= \left[f( x) \right]+ \left\{ f(x) \right\} $, where $ \left\{ x \right\} $ is the fractional part function, one can write by a direct substitution for the function $M(x)=\...
2
votes
1answer
39 views

$T$ can be $\infty$ with positive probability

From Williams' Probability with Martingales How exactly do we know $T$ can be $\infty$ with positive probability or $$P(T = \infty) > 0 \text{ ?}$$ I'm guessing that that means there ...
1
vote
4answers
27 views

Sum of $1/n^k$ of the first $\log P$ numbers

In a Udacity course I'm told the following: $\sum_{i=1}^{\log_2 (P)} 1/2^i = (P-1) /P $ I've checked that it's true by entering it into Wolfram Alpha: https://www.wolframalpha.com/input/?i=sum+1%2F2^...
1
vote
1answer
51 views

Evaluate $\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$

Find a closed form expression for $$\sum_{r=1}^{\infty} \dfrac{\sin(r\pi x)}{r \cdot y^r}$$ I know that $\displaystyle\sum_{r=1}^{\infty} \dfrac{\sin(r \pi x)}{r} = \dfrac{\pi}{2} - \...
1
vote
1answer
36 views

A Function of a Convolution (Laplace)

A paper I am reading makes the following claim: Assume that $a_n$ is a series of of positive, distinct, real numbers. Assume that $\epsilon_n$ are independent random standard exponential variables. ...