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Questions tagged [summation]

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

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8 views

Maximization of coupled circulants sum wrt some constraints

I'm trying to maximize the following expression: $$ \begin{align}\sum_{i = 0}^n a_i \cdot \sum_{j=0}^n s_j \cdot t_{ (i + j) \bmod n}\end{align}$$ With the constraints: $ 0 \le t_z, s_z \le Q$ $ \...
2
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3answers
54 views

Sum of certain consecutive numbers is $1000$.

Question: The sum of a certain number (say $n$) of consecutive positive integers is $1000$. Find these integers. I have no idea how to approach the problem. I did try the following but did not ...
0
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1answer
30 views

Compute the sum $ \sum_{n=1}^{\infty}(\frac{2}{3})^{n-1}\frac{1}{3}e^{-\frac{2n(1-t)}{t}}$

As part of a probability exam, we were required to compute the following sum: $ S=\sum_{n=1}^{\infty}(\frac{2}{3})^{n-1}\frac{1}{3}e^{-\frac{2n(1-t)}{t}}$ Our lecturer likes finding probabilistic ...
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3answers
36 views

Sum of increasing powers

In the answer to a different question someone wrote: Let $\omega = e^{2 \pi i / n}$ which implies $\omega^n = 1$. $$ 1 + \omega + \omega^2 + \ldots + \omega^{n-1} = \frac{\omega^n-1}{\omega-1} = ...
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1answer
48 views

Compute: $\frac3{7\cdot2}+\frac3{7\cdot12}+\dots$

Compute: $$\frac3{7\cdot2}+\frac3{7\cdot12}+\frac3{17\cdot12}+\dots+\frac3{2017\cdot2012}$$ I couldn't really find the pattern in this one. I tried evaluating the first two terms which was $\frac14$, ...
0
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3answers
35 views

Evaluate $\sum_{k=1}^n\log(x ^ {\frac{k}{2k+1}} )$ [on hold]

I have absolutely no idea on how to tackle this. The log is completely throwing me off, so if you could explain how to deal with a log in a series i would really appreciate it!
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2answers
66 views

Do these polynomials with harmonic number-related coefficients lie in some particular known class?

I've generated a set of univariate polynomials ($b=1,2,\ldots$) in $v$ of degree $b-1$. The constant term and the coefficient of $v^{b-1}$ is simply $H_b$, the $b$-th harmonic number. The ...
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0answers
26 views

Upper and lower bounds for a finite sum [duplicate]

Find upper and lower bound for the following finite sum: $\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3}$ My attempt: Using the integral test: we know that $\frac{1}{1} + \frac{...
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0answers
32 views

Upper and Lower bound of a finite sum

Find upper and lower bound for the following finite sum: $$\frac{1}{1} + \frac{1}{2^3} + \frac{1}{3^3} + ··· + \frac{1}{n^3} $$ My attempt is: Using the integral test: we know that $\frac{1}{1} + ...
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1answer
57 views

find the upper and lower bound for a finite sum

Find upper and lower bound for the following finite sum $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ My attempt: $1/(1 + 1^3)+1/(1 +2^3)+1/(1 + 3^3) + ··· + 1/(1 + n^3)$ = $\sum_{i=1}^n ...
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1answer
38 views

Evaluating a sum (alternating binomial series with odd denominators) [duplicate]

How do I evaluate the following sum (for some positive integer $m$)?: $$S=\sum_{k=0}^{m}{{m \choose k}\frac{(-1)^k}{2k+1}}$$ After expanding it looks like: $$S={m \choose 0}-\frac{1}{3}{m \choose 1} + ...
3
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0answers
26 views

Finite double sum $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$; any advanced summation technique?

Let $M,N,c$ be positive integer. It was astonishing when trying to solve $\sum_{k=0}^N\sum_{l=0}^M\left\lfloor\frac{k+l}{c}\right\rfloor$ to obtain this rather complex looking result \begin{align*} ...
8
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1answer
48 views

If integration is a continuous analog of summation (Addition), what is the continuous analog of multiplication (Product)?

One definition of integration over a continuous interval [a,b] into n subintervals with equal width $\Delta x$, and from each interval choose a point $x_i^*$. Then the definite integral of $f(x)$ ...
1
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1answer
37 views

Estimate for multiple harmonic sum

I am interested in estimating the following family of sums: $$S_k(n) \equiv \sum_{\substack{n_1, \ldots, n_k \geq 1\\n_1 + \ldots + n_k = n}}\frac{1}{n_1\ldots n_k}$$ where $k \geq 1, n \geq 1$. A ...
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0answers
17 views

Understanding the intuition behind the direct proof of the closed form expression for the sum S of the first n positive integers.

I understand the proof by induction, however I would like to understand the intuition behind why when doing the direct proof we simply layout the the sum forwards and backwards and add the two as such:...
3
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2answers
59 views

Binomial sum having positive and negative terms

Find $\displaystyle \binom{n}{0}-\binom{n}{1}\frac{1}{4}+\binom{n}{2}\frac{1}{7}+\cdots \cdots $ What I tried: the sum is $$\sum^{n}_{r=0}(-1)^r\binom{n}{r}\frac{1}{3r+1}$$ $$\sum^{n}_{r=0}(-1)^r\...
-2
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1answer
31 views

Is the closed form of $\sum_{i=1}^n ni$ equal to $n\cdot\frac{n(n+1)}{2}$? [on hold]

I think I found the closed form of this summation $$\sum_{i=1}^n ni$$ but I am not sure. I think the closed form is $$n\cdot\frac{n(n+1)}{2}$$
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2answers
145 views

Proof by induction with sqaure root in denominator: $\frac1{2\sqrt1}+\frac1{3\sqrt2}+\dots+ \frac1{(n+1)\sqrt n} < 2-\frac2{\sqrt{(n+1)}}$

I need to prove $\frac{1}{2\sqrt1} + \frac{1}{3\sqrt2} + ... + \frac{1}{(n+1)\sqrt n} < 2 - \frac{2}{\sqrt{(n+1)}}$ by induction for every $n \in \mathbb{N} $. Please help, I am stuck for 2 days ...
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0answers
47 views

Simplifying $\frac{\sum^{n}_{k=1}\sec^2\theta_k}{\sum^{n}_{k=1}\csc^2\theta_k}$, where $\theta_{k}=\frac{2^{k-1}\pi}{2^n+1}$

If $\displaystyle\theta_{k}=\frac{2^{k-1}\pi}{2^n+1}$ and $\displaystyle a_{k}=\sec^2(\theta_{k})$ and $\displaystyle b_{k}=\csc^2(\theta_{k}).$ Then value of $$\frac{\sum^{n}_{k=1}a_{k}}{\sum^{n}_{k=...
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2answers
36 views

Sum of #s on dice.

Six standard six-sided dice are rolled, and the sum $S$ is calculated. What is the probability that $S × (42 – S ) < 297?$ Express your answer as a common fraction. First off can I ONLY just have ...
1
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2answers
63 views

How to convert the following sum to a geometric series?

Find $$ \sum_{n = 1}^{\infty} \frac{6 - 2^{2n - 1}}{3^n} $$ There are many ways to find that the limit is divergent, but the question explicitly states the sum must be interpreted as a geometric ...
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0answers
19 views

Absolute oscillator in Langton's Ant

We have a simple (or single) block of Langton's Ants colony which includes two ants looking in the same direction. Their positions can be interpreted as knight's walk. The distances between each next ...
4
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1answer
66 views

Simplifying $\frac{{\sum_{i=1}^{i=n}}1+\tan^2\theta_i}{{\sum_{i=1}^{i=n}}1+\cot^2\theta_i}$, where $\theta_i = \frac{2^{i-1}\pi}{2^n+1}$

How to simplify this expression? $$\frac{\sum_{i=1}^{n}\left(1+\tan^2\theta_i\right)}{\sum_{i=1}^{n}\left(1+\cot^2\theta_i\right)}$$ where $$\theta_i = \frac{2^{i-1}\pi}{2^n+1}$$
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2answers
46 views

Infinite sum of $\frac{x^n}{n+1}$

I was solving a calculus problem and found myself stuck trying to find to solve: $$S = \sum_{n=0}^{\infty} \frac{x^n}{n+1}$$ Wolfram Alpha says: But I have no idea how this sum is evaluated. Any ...
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2answers
26 views

How do i prove this problem with a product and its fraction part? [closed]

How do I prove that $\forall \:\:r\:\in \mathbb{R}$,$\:r\:>\:1,\:\sum _{i=0}^n\:xr^i=x\frac{1-r^{n+1}}{1-r}$.
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1answer
15 views

Using a Euclidean norm to bound a $k$-tuple

This does not look too complicated, but I've been stuck here for several hours. My question is to prove that $||(h, \cdots, h)||\leq ||h||^{k}$, where $||\cdot||$ is the euclidean norm, and $(h,\cdots,...
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0answers
21 views

How to sum over all allowed values of adjacency matrix?

I have troubles to understand the following simplification. $\sum_{\{A_{ij}\}} \Pi_{i<j} e^{\lambda A_{ij}} = \Pi_{i<j} \sum_{A_{ij =0,1}} e^{\lambda A_{ij}} $ How am I allowed to move the ...
2
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1answer
48 views

Find the value of $S_1+S_2$

Knowing that $$\sum_{k=1}^{\infty}\frac{1}{k^2}=\frac{\pi^2}{6}$$ and $$S_i=\sum_{k=1}^{\infty}\frac{i}{(36k^2-1)^i}$$ Find value of $S_1+S_2$ i tried splitting: $$\frac{1}{36k^2-1}=\frac{1}{2}\...
3
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2answers
41 views

A sum of series problem with alternating sign of terms

I came across a problem that requires me to find the sum of a series. The term of the series $T_n$ is given by $$T_n = (-1)^{\frac{n(n+1)}2}n^2$$ Sum till $4n$ terms is to be found. Writing down the ...
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0answers
22 views

Harmonic series (double)

I am wondering about the $\Theta$ class (i.e. asymptotic complexity) of the following: $$\sum^n_{i=1} \sum_{j=1}^n \frac{1}{ij}$$ Since this is basically the harmonic series, applied twice, it ...
4
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1answer
96 views

Symmetry in function given by double sum

I had to deal with this function: $$ f_n(x_1,x_2)=(x_2-x_1)^{n-1}\sum_{m=0}^{n-1}\sum_{j=0}^{n-m-1}C(n,m,j)\left(\frac{x_2}{x_2-x_1}\right)^m\left(\frac{x_2(1-x_1)}{x_2-x_1}\right)^j $$ where $$C(n,...
6
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1answer
84 views

Is there a sum of an uncountable set of Real numbers? [duplicate]

The addition of Real numbers is commutative, so instead of saying we can find the sum of a sequence $\{a_1,...,a_n\}$ of real numbers that are pairwise not equal, we can say that there is a sum of a ...
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0answers
20 views

What is the limit of the series (summation) of the q-Pochhammer symbol or the ~q-Pochhammer symbol?

I am interested in knowing if the following series converges or not: \begin{equation} \sum_{n=1}^{\infty} \prod_{i=1}^n \left(1-e^{-\sqrt{i}} \right) \qquad Expression \; 1 \end{equation} If that is ...
1
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1answer
48 views

Fast formula for $\sum_{i=1}^{n} (i \cdot (i!))$, for arbitrary $n \in \mathbb{N}_1$

I've been reading How To Prove It second edition by Daniel J. Velleman, and I've encountered an end-of-subsection exercise I can't answer. On page 286, exercise 10 of subsection 6.3 states: "Find a ...
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1answer
19 views

Finding a quantity between two points in three dimensional space

I'm obligated to let you know I've cross posted this on stack overflow earlier today but decided after some comments that I this question is probably less one of code and more one of the mathematics ...
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1answer
16 views

Summation Rules Mistake?

I'm looking at a solution for a stats problem and I think there is a mistake. The solution is as follows: $$m_i^{-1} \sum_{e=1}^{m_i} u_{i,e} = m_i^{-1}\sum_{e=1}^{m_i}(f_i + u_{i,e}) = f_i + m_{i}^{-...
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3answers
43 views

Evaluate $\sum ^n _{r=0} \binom{n}{r} \tan^{2r}\left(\frac \pi 3 \right)$

Evaluate $$\sum ^n _{r=0} \binom{n}{r} \tan^{2r}\left(\frac \pi 3 \right)$$ So I've got to a point at which I don't know how to go any further, any help would be appreciated. My workings so far are ...
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0answers
38 views

Sum of natural numbers? [duplicate]

Is the sum of all natural numbers equal to infinity or -1/12. I have seen convincing evidence on the Internet that the solution is -1/12. But natural numbers are all positive (unless we consider zero),...
1
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1answer
23 views

Writing down an arbitrary curve in a formal way

I want to write down a curve, which I define as the set of points $x,y\in\mathbb{F}$ (a field) such that $f(x,y)=0$, but I write it as the polynomial $f$ and call $f$ the curve. So I would like to ...
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2answers
52 views

Why $ \sum_{k=0}^{\infty} \frac{n^k}{k!} = e^n$? [closed]

A lecture note used the following claim: $$ \sum_{k=0}^{\infty} \frac{n^k}{k!} = e^n$$ Why is that?
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1answer
27 views

Upper and lower bounds for series and sequences [on hold]

Find upper and lower bounds for the following finite sum: $$1 + \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{3}} + \cdots + \frac{1}{\sqrt{n}}$$
4
votes
2answers
102 views

show this inequality $a_{1}+a_{2}+\cdots+a_{n}\ge\sqrt{3}(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n}a_{1})$

let $a_{1},a_{2},\cdots,a_{n}\ge 0,n\ge 3$,and such $$a^2_{1}+a^2_{2}+\cdots+a^2_{n}=1$$ show that $$a_{1}+a_{2}+\cdots+a_{n}\ge\sqrt{3}(a_{1}a_{2}+a_{2}a_{3}+\cdots+a_{n}a_{1})$$ I can prove ...
3
votes
3answers
61 views

Proving an identity involving the alternating sum of products of binomial coefficients

Prove the following identity: $$ \sum_{k=0}^{l}(-1)^k \binom{j-k}{l-1}\binom{l}{k}=0$$ for some integers $l\geq1$ and $j\geq l$. Using wolfram alpha I have confirmed that this identity is true. But ...
2
votes
1answer
46 views

The limit of the maximum of a sum of sines

I've recently stumbled upon the following problem from Brilliant: Compute the following: $$\lim_{n\to\infty}\max_{x\in[0,\pi]}\sum_{k=1}^n\frac{\sin(kx)}k$$ Options: $\...
0
votes
3answers
44 views

A lower bound for de Polignac's formula

De Polignac's Formula has many uses, for example when calculating the number of trailing zeroes of $n!$ :$$\nu_5(n!)=\sum_{i\le\lfloor\log_5n\rfloor}\left\lfloor\frac n{5^i}\right\rfloor.$$ For the ...
6
votes
4answers
929 views

Sigma notation for sum of $\ln(x)^2$ from $2$ to $20$ with steps of $0.5$

Is it possible to use sigma notation for non-integer steps, for example I want to sum $\ln(x)^2$ from $2$ to $20$ with steps of $0.5$, is there a way I could write this in sigma notation or some other ...
0
votes
3answers
65 views

If $\sum_{n=1}^\infty\tan^{-1}\left(\frac4{n^2+n+16}\right)=\tan^{-1}\left(\frac\alpha{ 10 }\right)$, then find $\alpha$.

$$\sum _ { n = 1 } ^ { \infty } \tan ^ { -1 } \left( \frac { 4 } { n ^ { 2 } + n + 16 } \right)= \tan ^ { -1 } \left( \frac { \alpha } { 10 } \right)$$ Find $\alpha$. I know I need to convert to $$\...
1
vote
3answers
51 views

Evaluate $\sum\limits_{k=1}^n (k^{3} +k^{2} +1) / (k^{2} +k)$

I'm a beginner at summations, and my first instinct for this sum was to use a partial fraction. This didn't really work even after I tried factoring the polynomials, i think because the numerator has ...
0
votes
3answers
42 views

How do you compute the sum of k * a^k

We have the sum $$\sum_{k=0}^{n} a^k k,$$ where a is a constant and we need the answer in terms of $n$. How can we go about solving this? If $a$ were a variable we could use differentiation with $\...
1
vote
1answer
48 views

If $\sum_{i=1}^n \cos \theta_i$=n, then find the value of $\sum_{i=1}^n \sin \theta_i$ [closed]

If $\displaystyle\sum_{i=1}^{n} \cos(\theta_{i}) = n$, then find the value of $\displaystyle\sum_{i=1}^n \sin(\theta_i)$