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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.

1
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0answers
38 views

When is $a(n)$ prime?

Question: When is $a(n)\in P$ compared to all possible values of $n$? where $P$ denotes the set of primes. What is the density of the primes in the sequence? Consider the sum of the prime counting ...
1
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0answers
54 views

Approximating an infinite sum: $\sum_t t^d \exp(-(t-1)^2/2)$

I am seeking to upper bound the limit of the following infinite summation, when a free parameter $\beta$ can be chosen, perhaps dependent on $d$, to help reduce the sum: \begin{equation} f(\beta,d) = \...
0
votes
3answers
40 views

Prove that $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \frac{e^x - e^{-x}}{2}$

I have been trying to show: $\sum_{n=0}^\infty {\frac{x^{2n+1}}{(2n+1)!}} = \left(\frac{e^x - e^{-x}}{2} \right)$ I have come so far as to show: $\begin{aligned} \sum_{n=0}^{\infty} {\frac{x^{2n+1}}...
-2
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0answers
20 views

swapping the order of summation [on hold]

https://photos.app.goo.gl/qgVNDcGe9GbRycCa6 [enter link description here][1]
2
votes
2answers
61 views

Sum of the sum of the sum of $4i(-1)^i$

I am attempting to evaluate the sum $$ \sum_{i=0}^{x-1} \Biggl(4+4\sum_{j=3}^{i+2} j(-1)^j\Biggr) $$ I'm not that experienced at summations, so I can go as far as $$ 4x+4\sum_{i=0}^{x-1} \sum_{j=3}^{...
14
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3answers
279 views

Sum of $\{n\sqrt{2}\}$

I would like to prove (rigorously, not intuitively) that $$\sum_{n=1}^N \{n\sqrt{2}\}=\frac{N}{2}+\mathcal{O}(\sqrt{N})$$ where $\{\}$ is the "fractional part" function. I understand intuitively why ...
0
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2answers
24 views

Why is $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\delta(k-N+k_0)$?

EDIT: $\delta$ is the Dirac delta function and in the context it is defined as $\delta(0)=1$ and $0$ for all $n\neq 0$. I am having trouble concluding that $\sum_{n=0}^{N-1}e^{-2i\pi(k+k_0)n/N}=N\...
-2
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0answers
38 views

Decimal digit extraction of $\pi$

I've seen many folks asking for this so I thought I'd take a shot @ answering it: This function $\pi[d]:\mathbb{N}\rightarrow\left\{\mathbb{W}<10\right\}$, based on the BBP closed form expression, ...
3
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2answers
265 views

$\sum_{k=-m}^{n} \binom{m+k}{r} \binom{n-k}{s} =\binom{m+n+1}{r+s+1}$ using Counting argument

I saw this question here:- Combinatorial sum identity for a choose function This looks so much like a vandermonde identity, I know we can give a counting argument for Vandermonde. However much I try ...
0
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0answers
9 views

Normal Equations (OLS) : Summation notation to system of equation form

I am struggling to "visualize" the following equation. I would like to know how the following equation can be written in system-of-equation form in order to see that there are $K$ unknowns. The ...
1
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2answers
36 views

Sum $\sum_{i = 1}^N \sum_{j = i + 1}^N \mathbb{I} (j - i \leqslant K)$

Let $N,K$ be non-negative integers. What's the value of the following sum? $$S(N,K) = \sum_{i = 1}^N \sum_{j = i + 1}^N \mathbb{I} (j - i \leqslant K)$$ where $\mathbb{I}(\mathcal P)=1$ if $\mathcal ...
6
votes
1answer
79 views

Find $\lim_{t\to 1^{-}}(1-t)\sum_{r = 1}^\infty \frac{t^r}{t^r+1}$

$$\lim_{t\to 1^{-}}(1-t)\sum_{r = 1}^\infty \frac{t^r}{t^r+1}$$ Note: I am a high school student and this problem appeared in my test. So, please try to use methods to solve this problem at a high ...
0
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1answer
40 views

What constant $c$ will make $ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N ?$

What constant $c$ will make this equality valid for any $N$ chosen? $$ \sum_{k=2}^{N}c^{\frac{1}{k\log k}}=N. $$ I tried getting a rough idea of what $c$ should be and got about $1.46$ when $N=1000$ ...
1
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3answers
40 views

Summation formula for this?

I have found the following summation formula based on a recurrence. It supposes $n = 2^k$ where k is an integer. I've intuitively discovered that the following closed form may be true (following the ...
4
votes
4answers
209 views

Closed form of recurrent arithmetic series summation

Knowing that $$\sum_{i=1}^n i = \frac{n(n+1)}{2}$$ how can I get closed form formula for $$\sum_{i=1}^n \sum_{j=1}^i j$$ or $$\sum_{i=1}^n \sum_{j=1}^i \sum_{k=1}^j k$$ or any x times neasted ...
6
votes
3answers
288 views

What is the value of $\frac11+\frac13-\frac15-\frac17+\frac19+\frac1{11}-\dots$?

The series $\sum_{k=1}^{\infty }\frac{(-1)^{k+1}}{2k-1}=\frac{1}{1}-\frac{1}{3}+\frac{1}{5}-\frac{1}{7}+\dots$ converges to $\frac{\pi}{4}$. Here, the sign alternates every term. The series $\sum_{k=...
1
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2answers
48 views

Sum of alternating binomial-coefficient-type series

Let $D,n\in \mathbb N$ with $0<D<n$, and $y>0$ is a real number. Question: Is there a closed-form for the following alternating sequence \begin{equation} \sum_{k=0}^D (-y)^k {n\choose k}? \...
0
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1answer
11 views

Algorithm Complexity - Summation - Correctly interpreted how to do it.

I wanted to double check my understanding and working out for a 3 nested for loop algorithm, and working out it's complexity. I've got the right answer, but how I've arrived at it I feel isn't exactly ...
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0answers
33 views

Help on converting summation to integration [on hold]

Can someone help me on turning the sum $$\sum_{i=1}^\infty 1+\frac{i\lfloor\frac{x}{i}\rfloor-x+\frac{1}{2}}{\vert i\lfloor\frac{x}{i}\rfloor-x+\frac{1}{2}\vert}$$ into an integral, I have tried but ...
-1
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1answer
61 views

$\sum_{n=1}^\infty \frac{n}{4n^4+1}$ converges to? [on hold]

$$\sum_{n=1}^\infty \frac{n}{4n^4+1}$$ my attempt : assumed the series is a telescopic and tried finding $t_n - t_{n-1}$ but then realized it is not a telescopic series. $$$$ //answer is given to be 0....
0
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0answers
30 views

Separating out the last term of a given sequence

I am suppose to separate the last term of the given sequence. $$\sum_{j=0}^n2^j$$ my work as shown... $$ \sum_{j=0}^{n-1}2^j\ +\ 2^n$$ Does this appear wrong?
0
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1answer
38 views

Infinite sums and squares

So, I'm sure this has been thought of and said before but I'm curious. So $\sum \frac{1}{2^n}$ can be thought of by filling up a square. First we color in a whole square and then we draw a second ...
1
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4answers
53 views

Find the value of $\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$

Find the value of $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ My apporach:- $$\sum_{r=1}^4 \log_2 (\sin(\frac{r\pi}{5}))$$ $$=\log_2 (\sin(36^{\circ}))+\log_2 (\sin(2*36^{\circ}))+\log_2 (\sin(3*...
-2
votes
1answer
49 views

How to convert an infinite summation to an integral [on hold]

I need help converting this summation to an integral $$\sum_{i=1}^\infty 1+\frac{\vert(n-im)\vert}{(n-im)}$$ I keep trying but get stuck so any help is appreciated.
1
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1answer
69 views

Find the infinite sum $\sum^\infty_{n=2}\frac{\left(-1\right)^{n}}{\log n}$, speed up its convergence

$\displaystyle\sum_{n\in\mathbb{Z}_{\geqslant{2}}}\frac{\left(-1\right)^{n}}{\log n}$ Since $\log x$ grows more slowly than any positive power, the sum above converges extremely slowly. Is there a ...
1
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0answers
21 views

Rewriting a set of integers to get rid of repetition but keeping subset sum ordering

Say, I have a set of 6 +ve integers sorted in ascending order: $A = \{2,4,4,4,5,7\}$ Now to make it easier to deal with (Minimum one starts with 1) I deducted one from all of them: $\therefore B= ...
3
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2answers
85 views

Derive the sum of $\sum_{i=1}^n ix^{i-1}$

For the series $$1 + 2x + 3x^2 + 4x^3 + 5x^4 + ... + nx^{n-1}+... $$ and $x \ne 1, |x| < 1$. I need to find partial sums and finally, the sum $S_n$ of series. Here is what I've tried: We ...
0
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0answers
14 views

Proving $\sum_{j=0}^{N-1}\cos\frac{\left(2j+1\right)\pi}{2N}=0$ [duplicate]

Let $l\in\mathbb{Z}$ and $N\in\mathbb{N}$. I need to prove the following: \begin{equation} \sum_{j=0}^{N-1}\cos\left(l\frac{\left(2j+1\right)\pi}{2N} \right)=0 \end{equation} I tried to use Euler ...
0
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1answer
42 views

Find the Sum of the Series: $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$

Find the Sum of the Series $$\sum_{n=0}^{\infty}\frac{(-1)^n\pi^{2n}}{6^{2n}(2n)!}$$ Alright, so I think I may have gotten this problem correct but I'm a little hesitant, so If you could check my ...
3
votes
2answers
74 views

Find the partial sum formula of $\sum_{i=1}^n \frac{x^{2^{i-1}}}{1-x^{2^i}}$

Given next series: $$\frac{x}{1 - x^2} + \frac{x^2}{1 - x^4} + \frac{x^4}{1 - x^8} + \frac{x^8}{1 - x^{16}} + \frac{x^{16}}{1 - x^{32}} + ... $$ and $|x| < 1$. Need to derive $S_n$ formula ...
0
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0answers
62 views

What is $\sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$?

The equations $$ \sum_{k=1}^m\begin{pmatrix} n \\ k \end{pmatrix}^2 \quad \text{and} \quad \sum_{k=1}^m\left[2^k\begin{pmatrix} n \\ k \end{pmatrix}\right]^2$$ popped up in some of my calculations, ...
0
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0answers
25 views

Is it incorrect to have a sum of an infinite weighted set?

I am currently in a revision of a paper. I have found something that I would only like to change if it is currently really notationally false because each change bears a few risks. What I currently ...
-1
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1answer
42 views

Sum of sequential numbers rational or irrational? [closed]

Is following infinite sum is rational or irrational? $\sum_{n=1}^\infty n/10^n$
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2answers
35 views

$\sum_{n=1}^{\infty}\sum_{i=nr}^{\infty}f(i)=\sum_{i=r}^{\infty}\sum_{n=1}^{\left\lfloor \frac{i}{r}\right\rfloor}f(i)$

This was given as part of the answer of a more complex problem: $$\sum_{n=1}^{\infty}\sum_{i=nr}^{\infty}f(i)=\sum_{i=r}^{\infty}\sum_{n=1}^{\left\lfloor \frac{i}{r}\right\rfloor}f(i)=\sum_{i=r}^{\...
1
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2answers
41 views

Complex series involving hyperbolic cosine

Please how to calculate the sum of such series! I need the idea ! $$\sum _{n=1}^{\infty} \cosh(n)\frac{z^{2n}}{n!} $$ $$\sum _ {n=0}^{\infty} \frac{(1+i)^{n}z^{n}}{n!}$$
0
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3answers
30 views

Evaluate $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n+1}}{3^{2n}(2n+1)!}$

Evaluate $\sum_{0}^{\infty}\frac{(-1)^n\pi^{2n+1}}{3^{2n}(2n+1)!}$ For this homework problem, I really don't have any clue how to start it, so any hints are welcome. But my first intuition would be ...
0
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2answers
25 views

Exponential double angle formula

My question is whether someone could provide a proof for the following identity: $$ \frac{1 - e^{int}}{1 - e^{it}} = e^{i(n-1)t/2} \frac{\sin(nt/2)}{\sin(t/2)} $$ Motivation: The left hand side is ...
0
votes
1answer
29 views

Find the Taylor series of $\frac{1+z}{1-z}$ at $z_{0}=i$

I'm given the following explanation: Let $\dfrac{1+z}{1-z} = \biggl(\dfrac{1+i}{1-i}+\dfrac{z-i}{1-i}\biggr)\biggl(1-\dfrac{z-1}{1-i}\biggr)^{-1}$ = = $\dfrac{1+i}{1-i}\displaystyle\sum_{j=0}^{\...
0
votes
2answers
33 views

Find the value of $a$ such that the series converges.

I have to find the value of a such that the following series converges:$$\sum_{n=1}^{\infty}n^{\frac{1}{3}}\left|\sin\left(\frac{1}{n}\right)-\frac{1}{n^a}\right|$$ First of all, I simplified the ...
1
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3answers
94 views

Prove that $4 + 5 + 6 + \cdots+ n = \frac{n(n+1)}{3}$ for $n \ge 4$

Prove using mathematical induction that $$4 + 5 + 6 + \cdots + n = \frac{n(n+1)}{3}$$ where $n \geq 4$ is an integer. I just wanted to confirm because my Base case P(4) is false. So this statement ...
10
votes
3answers
99 views

Prove that $\lim_{a \to \infty} \sum_{n=1}^{\infty} \frac{(n!)^a}{n^{an}} = 1$.

The above sum (without the $\lim$ notation) is convergent $\forall a \in \Bbb{N}^+$, because: $$ \sum_{n=1}^{\infty} \frac{(n!)^a}{n^{an}} = \sum_{n=1}^{\infty} \frac{n! n! \dots n!}{n^n n^n \dots n^n}...
2
votes
2answers
102 views

Find $\lim _{n\rightarrow \infty }\frac {na^{2}_{n}+1}{\sum ^{n}_{k=1}\left( 1+2+3+\ldots +k\right) }$

Define $\left\{ a_{n}\right\} $ is an arithmetic sequence that all terms are positive integers. If $a_{10}-a_{1}=225$, find $$\lim _{n\rightarrow \infty }\dfrac {na^{2}_{n}+1}{\sum\limits^{n}_{k=1}\...
1
vote
1answer
2k views

Formula to find the sum of nth row?

In the following triangle I need to find the sum of nth row. Is there a general formula for this? If yes, then please tell me. Triangle: Row 1: 1 Row 2: 1 2 1 Row 3: 1 3 6 3 1 Row 4:...
2
votes
2answers
69 views

Evaluating $\sum_{k=1}^{\infty}\frac{1}{k(3k-1)}$

I am wondering if the sum $$S=\sum_{k=1}^{\infty}\frac{1}{k(3k-1)}$$ has an exact expression. And when I plugged it into Wolfram Alpha it spitted out: $$S=\frac{1}{6}\Big(-\sqrt{3} π + 9 \ln(3)\Big)$$...
0
votes
3answers
46 views

Why is $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?

Assume we have the sum $\sum\limits_{i = 1}^n \frac{n}{n-i+1}$ why is this equal to $n \sum\limits_{i = 1}^n \frac{1}{i}$?
0
votes
3answers
45 views

Evaluate a sum involving the Fibonacci sequence [closed]

Evaluate the sum $ \displaystyle \sum_{i=0}^{\infty }\frac{f_{i}}{2^{i}},$ where $f_{n}$ is the n-th term in the Fibonacci sequence.
2
votes
2answers
55 views

To find the sum: $\frac {1}{n!} \sum \binom {n}{2+3r} x^{1+r}$

Sum the series: $$ \frac {x}{2!(n-2)!}+\frac {x^2}{5!(n-5)!}+\frac {x^3}{8!(n-8)!}+....+\frac {x^{\frac {n}{3}}}{(n-1)!}, $$ $n$ being a multiple of $3$.(Math. Tripos, 1899) My attempt We may ...
0
votes
1answer
21 views

Unsure how the following summation simplifies down to this known result?

How does this: $$\frac{1}{c+1} + (c-1)\cdot\left(\frac{1}{c(c+2)}+\dotsb+\frac{1}{(n-2)n}\right) + \left(\frac{c-1}{n-1}\right)\cdot\frac{1}{2}$$ Become this: $$= \frac{2cn-c^2+c-n}{cn}$$ PS. $1 \...
2
votes
5answers
53 views

Series convergence of $\sum_{n=1}^{\infty}\frac{1}{n^3}$

I have the infinte series $\sum_{n=1}^{\infty}\frac{1}{n^3}$ which I believe converges. As the ratio test proved inconclusive, I am trying to use the comparison test in order to prove it's ...
0
votes
1answer
31 views

How to Solve a “Sum Inverse” [closed]

Before I start, I want to say that the kinds of problems like (solve for $x$: $1=\sum_{n=1}^x(\ln(n))$) is not what I'm talking about. I'm talking about if you have a certain function like $\ln(x)$ ...