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

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2
votes
0answers
11 views

Is it possible to eliminate the inner sum to evaluate numerically?

Any hints on how to simplify the following double sum to be able to find the sum at least numerically? $$\sum_{n=2}^{\infty}\frac1{n(n^2-1)} \sum_{k=0}^\infty ...
0
votes
4answers
34 views

How to sum $\sum_{r=1}^n q^{(r-1)k}$?

I need to find $$\sum_{r=1}^n q^{(r-1)k}$$ but I'm unsure on how to do this. Any help? I believe it may be something to do with a geometric progression but I'm not sure what to do with this. Thanks.
5
votes
0answers
36 views

Evaluate $S=\left|\sum_{n=1}^{\infty} \frac{\sin n}{i^n \cdot n}\right|$

Evaluate $$ S=\left|\sum_{n=1}^{\infty} \dfrac{\sin n}{i^n \cdot n}\right|$$ where $i=\sqrt{-1}$ For this question, I did the following, Let $$ \begin{align*} S &= \sum_{n=1}^{\infty} ...
0
votes
1answer
19 views

How to compute $\frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} w^{n^2} = \frac{1}{2}(1+i)\left(1+(-1)^N\right)$? [on hold]

For $w=\exp[\frac{2\pi i}{N}]$, (i.e $N$th root of $1$.) \begin{align} \frac{1}{\sqrt{N}} \sum_{n=0}^{N-1} w^{n^2} = \frac{1}{2}(1+i)\left(1+(-1)^N\right) \end{align} How one can show this equation ...
2
votes
3answers
81 views

Simplify the sum $ \sum_{k=1}^{\infty} (\frac{1}{2})^kk $

I need some help simplifying this sum: $$ \sum_{k=1}^{\infty} \left(\frac{1}{2}\right)^kk $$ I have a feeling it's some basic series thing that I'm forgetting, but I need help nonetheless.
1
vote
2answers
36 views

How to evaluate the sum $\sum_{j = i+1}^{n-1} j $?

How would I go about solving this summation? $$\sum\limits_{j = i+1}^{n-1} j $$ I'm trying to figure out how to solve this summation using the fact that $\sum\limits_{i = 1}^{k} i= ...
3
votes
1answer
38 views

Name of Inequality

Let $x_i, y_i$ be complex numbers for all $i$. Is there a name for the following inequality? $$\left| \sum_{i=1}^n x_i \right| \leq \sum_{j=1}^n |x_j| $$ In particular, is it a special case of this ...
1
vote
0answers
12 views

Asymptotics of $\sum_{\mathfrak{a}}\frac{n^{k-\epsilon}}{\mathfrak{N}\left(\mathfrak{a}\right)^{r\left(k-\epsilon\right)}}$

In this paper by Brian D. Sittinger, the following claim is made: For an algebraic number field $K$ with norm $\mathfrak{N}$, let $\epsilon=\left[K:\mathbb{Q}\right]^{-1}$. Then, taking the sum over ...
3
votes
2answers
32 views

Proving $1+\sum_{i=1}^n i (i!)=(n+1)!$ [duplicate]

How would you prove the following using induction. n is a non negative integer $$1+\sum_{i=1}^n i i!=(n+1)!$$ This be what I did base case let $n=3$ $$1+1+4+18=(3+1)!$$ $24=24$ Hypothesis step ...
2
votes
1answer
54 views

closed form for a double sum

How can I prove that $$\underset{k\geq1}{\sum}\left(\underset{m=-\infty}{\overset{\infty}{\sum}}\frac{\left(-1\right)^{m}}{\left(2k-1\right)^{2}+m^{2}}\right)=\frac{\pi\log\left(2\right)}{8}\,?$$I ...
0
votes
1answer
21 views

Divisibility of binomial coefficients

I have got this series of binomial coefficients - $${2n\choose 0}+3\times{2n\choose 2}+3^{2}\times{2n\choose 4}+\ldots +3^{n}\times{2n\choose 2n}$$ I have to prove this to be divisble by ...
0
votes
0answers
44 views

Proving that an infinite sum of irrationals is irrational

First of all, I know this question may be closed because it is off topic, but I do have a valid question. Problem: Is is possible to prove that an infinite sum of distinct and different irrational ...
0
votes
1answer
14 views

Simplifying this summation

I've been doing this question and I'm stuck! Each customer who enters Larry’s clothing store will, independently of every other customer, purchase a suit with probability p. Assume that N, the ...
2
votes
3answers
52 views

Proof by induction, binomial coefficient

I have to make the following proof: $${\sum\limits_{k=1}^n}{k}{n\choose k} = n2^{n-1}$$ Base case, $n = 1$: $${\sum\limits_{k=1}^{1}}{k}{1\choose k} = 1 = 1\cdot2^0=1$$ Inductive Hypothesis: for ...
0
votes
2answers
18 views

Minimum Value Given Average

I am struggling with this (very embarrassingly basic) minimization problem: I have the answer, which is great, but I'm oblivious as to how to go about solving this short of using calculus. The ...
0
votes
0answers
5 views

how to calculate the composite score / ranking or weight

I have $14$ parameters ( for e.g $A, B, C, D$, etc) which I have obtained rating of all from $1$ to $14$. I have calculated their final weights using rank order centroid which are absolute weights ...
2
votes
1answer
17 views

Prove the series converges uniformly at $[x_0, \infty)$

Let $\sum_{n=0}^\infty a_ne^{-\lambda_n x}$, where $0 < \lambda_n < \lambda_{n+1}$. It is given that the series converges at $x_0$. Prove that the series converges uniformly at $[x_0,\infty)$. ...
6
votes
1answer
51 views

How to prove $\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$

I am trying to prove the following: $$\lim_{n \to \infty}\frac{\pi}{2n+1}\sum_{k=1}^{n}(-1)^{k+1}\cot\frac{k\pi}{2n+1}=\ln2$$ I tried some values and it seems convincing. I wonder if this is a ...
8
votes
1answer
167 views
+50

Identity in Number Theory Paper

In this paper by Jerry Hu, he defines the function $$f_{s,k,i}\left(u\right)=\prod_{p\mid u} \left(1-\frac{\sum_{m=i}^{k-1}{s \choose m}\left(p-1\right)^{k-1-m}}{\sum_{m=0}^{k-1}{s \choose ...
2
votes
2answers
59 views

Sum $\sum_{x=1}^n\sum _{y=1}^{x-1}\frac{1/2^x*1/2^y}{1/2^x+1/2^y}$

Is there a way to calculate following summation $\sum_{x=1}^n\sum _{y=1}^{x-1}\frac{1/2^x*1/2^y}{1/2^x+1/2^y}$ Can it be reduced to something simple?
1
vote
2answers
19 views

Understanding the logic behind this summation

The following is an excerpt from a proof that $\sum_1^n {i^k} = \theta(n^{k+1})$: $$\sum_1^n{i^k} \ge \sum_{\lceil n/2 \rceil}^n{i^k} \ge \sum_{\lceil n/2 \rceil}^n{\lceil n/2\rceil^k}$$ The first ...
0
votes
1answer
16 views

Maximum of sum of $k$-th powers with sum of bases equal to $n$

For some positive integer constants $n, k$ and $t$, I want to find the values for $n_1, \ldots, n_t$, all positive integers, that maximize the following sum : $$ \sum_{i = 1}^t (n_i)^k $$ such that ...
5
votes
5answers
71 views

How to show that $\sum_{k=1}^n k(n+1-k)=\binom{n+2}3$?

While thinking about another question I found out that this equality might be useful there: $$n\cdot 1 + (n-1)\cdot 2 + \dots + 2\cdot (n-1) + 1\cdot n = \frac{n(n+1)(n+2)}6$$ To rewrite it in a more ...
1
vote
5answers
48 views

How do you call this fact about sum of powers of n-th unity root?

I often see identity $$\sum_{k=0}^{n-1}e^{\tau ika/n} = \cases {n \quad \text{ if }n | a\\0\quad \text{ otherwise}}$$ in the context of generating functions. It allows to zero out all members of ...
5
votes
2answers
308 views

What is the proper notation for a general number of nested summations?

A sum over one index: $\sum_i f(i)$ A sum over two indices: $\sum_i \sum_j f(i,j)$ A sum over many indices: $\sum_{k_1} \sum_{k_2} \underbrace{\dots}_n \sum_{k_n} f(\mathbf k)$?
0
votes
1answer
25 views

Solving a summation where the inner summation is limited by the iterator of the two outer summations

I'm trying to solve the following summation (where C is some constant) but I'm stuck because of the inner most summation which is limited by $i\sqrt[2]{j}$ where i and j are the iterators of the outer ...
1
vote
7answers
49 views

Induction proof of $1 + 6 + 11 +\cdots + (5n-4)=n(5n-3)/2$

I need help getting started with this proof. Prove using mathematical induction. $$ 1 + 6 + 11 + \cdots + (5n-4)=n(5n-3)/2 $$ $$ n=1,2,3,... $$ I know for my basis step I need to set $n=1$ but I ...
0
votes
1answer
16 views

Alternative representation of time series

In a paper I am reading, it refers to the following time series model: $$ Y_t=\rho Y_{t-1}+e_t $$ Where $ \lvert\rho\rvert < 1$ It goes on to say that this process can be represented in the ...
1
vote
1answer
62 views

Proving an identity involving factorials

I have stumbled upon the following statement and have verified it computationally for many $n$ (up to n=500, it took a long time for my computer to do out all of the math), yet I have no idea how to ...
1
vote
1answer
30 views

Sum of continuous $L^{1}$ function over the integers.

Let $f$ be a continuous $L^1$ function defined on $\mathbb{R}$, such that $\hat{f}(k) = 0$ for $|k| > 1/2$, where $\hat{f}$ is the Fourier transform. Is it true that $\sum_{k = ...
1
vote
0answers
36 views

For the sum in the binomial theorem, what if the upper limit was different than n in the rest of the summand?

For the sum in the binomial theorem, what if the upper limit was different than n in the rest of the summand? Like this: $$ \sum_{k=0}^{m}\binom{n}{k}x^{k}y^{n-k} $$ Does it have a closed form at all? ...
-4
votes
0answers
31 views

Problem with Summation [on hold]

How would i apply the given formula and solve AP? could someone give me the numerical example (substation). I have tried doing the sum of Pt, +1 but the answer does not match the result in column ...
0
votes
2answers
15 views

What does the $i < j$ mean in $ E = -(\sum_{i<j} w_{ij}s_i s_j + \sum_i \theta_i s_i)$?

This is the energy function that is defined for Boltzmann Machines: $ E = -(\sum_{i<j} w_{ij}s_i s_j + \sum_i \theta_i s_i)$ What does the $i < j$ part mean for the running variables under the ...
0
votes
1answer
41 views

Where did the $-1$ come from?

It's a very specific question: Let $f(x) = \sum_{n=0}^\infty x^{n+2} = \frac{x^2}{1-x}$ $$f'(x) = \sum_{n=1}^\infty (n+2)x^{n-1} = \sum_{n=1}^\infty nx^{n-1} + 2\sum_{n=1}^\infty x^{n-1} = ...
2
votes
2answers
72 views

Evaluate $\sum_{n=1}^\infty nx^{n-1}$

How can you evaluate $\sum_{n=1}^\infty nx^{n-1} = \frac{1}{(1-x)^2}$ without relying on the fact that it's the derivative of $\sum_{n=1}^\infty x^n = \frac{1}{1-x} $?
0
votes
2answers
42 views

Generalized geometric series value

Why the value of the following summation: $$1 + \sum_{k=1}^{n}\bigg(1- \frac{76}{i}\bigg)^k= \frac{i}{76}$$ is $\frac{i}{76}$? $\quad i$ is a positive constant.
2
votes
0answers
27 views

What are conditions for an infinite sum with a complex parameter not to be analyitically extendable?

I'm looking for a sequence $f(n)$, so that $g(z):=\lim_{N\to\infty}\sum_{n=0}^N\exp\left(-z\cdot f(n)\right),$ with $z$ so that this converges classically, defines a function which can not be ...
2
votes
2answers
61 views

Geometric Series with coin tosses

Suppose you toss a coin and observe the sequence of H’s and T’s. Let N denote the number of tosses until you see “TH” for the first time. For example, for the sequence HTTTTHHTHT, we needed N = 6 ...
2
votes
1answer
20 views

Is there a way to express $(n-i)!(n-j)!(2i)!(2j)!$ in terms of $n$ and $r=i+j$?

I have been attempting to simplify the double sum: $$\sum_{i=0}^n \sum_{j=0}^n \frac{(-1)^{i+j} (2i+2j)!}{(n-i)!(n-j)!(2i)!(2j)!2^{i+j}(i+j)!}$$ And so what I am attempting to do is rewrite it in ...
1
vote
0answers
17 views

Techniques for computing (approximate or exact) partial sums for functions

Clearly there are several ways of computing the partial sum formulas of many summations, but is there a technique that can compute any partial sum. For example with $\sum_{x=0}^{n} \frac{1}{x}$, ...
-5
votes
2answers
54 views

Prove that $\sum_{k=0}^n q^k = \frac{1-q^n}{1-q}$ [closed]

Help me prove that: $$\sum^{n}_{k=0} q^k = \frac{1-q^n}{1-q}$$ and $$\sum^{\infty}_{k=0} x^k = \frac{1}{1-x}$$
20
votes
2answers
224 views

How to prove $\displaystyle\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$

How can we prove the following $$\sum_{n=0}^{\infty} \dfrac{1}{1+n^2} = \dfrac{\pi+1}{2}+\dfrac{\pi}{e^{2\pi}-1}$$ I tried using partial fraction and the famous result $$\sum_{n=0}^{\infty} ...
0
votes
1answer
15 views

Summation notation with ambiguous subscripts

I'm reading a paper which has the following description; Say we have a time series of correlated sequential observations of the random variable $X$ denoted $\{x_n\}_{n=1}^N$ from a stationary, time ...
1
vote
0answers
14 views

Summation of elements of a matrix in matrix notation

I have come across the following proof in a research paper. I feel the given end formula is wrong. Any help to correct it is greatly appreciated. where $V = \{v_{ij}\} \hspace{0.3cm} \text{with} ...
1
vote
2answers
65 views

Summation of an infinite series

The sum is as follows: $$ \sum_{n=1}^{\infty} n \left ( \frac{1}{6}\right ) \left ( \frac{5}{6} \right )^{n-1}\\ $$ This is how I started: $$ = \frac{1}{6}\sum_{n=1}^{\infty} n \left ( \frac{5}{6} ...
3
votes
1answer
38 views

$\sum_{i=0}^{k} \binom{m}{i}\binom{n}{k-i} =\binom{m+n}{k}$ [duplicate]

I'm trying to show that the equality $$\sum_{i=0}^{k} \binom{m}{i}\binom{n}{k-i} =\binom{m+n}{k}$$ Is true. I know it is since there is a good combinatorical argument for it. If we have a group of ...
1
vote
0answers
35 views

Partial sum formula for $\sum_{n=0}^{x} {\tan(x)}$ from $(0,\infty)$

I know there is no elementary way of expressing the partial sums of $\tan(x)$. I know; however, I can get an approximation of partial sums using a series, such as the MacLaurin series. If a series can ...
3
votes
4answers
100 views

Calculate $ S =\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}. $

Calculate $S =\displaystyle\sum_{k=1}^n\frac {1}{k(k+1)(k+2)}$. This sequence is neither arithmetic nor geometric. How can you solve this. Thanks!
2
votes
0answers
16 views

How to find a $\theta$ function verifying this property?

Let $r>4$ and $n>1$ be positive integers. Intuitively, the infinite sum $$S=\sum_{m=1}^{∞}\frac{2m}{r^{m^2}}$$ is related to a $\theta$ function. However, I cannot find a way to calculate this ...
2
votes
1answer
52 views

Can we simplify this sum?

Let $r>4$ and $n>1$ be positive integers. Can we simplify this sum: $$S=\sum_{m=1}^{n}\frac{2m}{r^{m^2}}$$ I have no idea to start.