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

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1answer
48 views

Proving Hermite's identity using induction

Can someone help me? This should be easy but I couldn't find it on any book or the internet. $$ \sum_{k=0}^{n-1}\left\lfloor x + \frac{k}{n}\right\rfloor = \lfloor nx \rfloor $$
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1answer
33 views

Frequency of Words in Document

I'm trying to figure this out: Would someone care to explain how one would go about using this function? More specifically, I don't understand the interval part, how does one count the intervals? ...
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2answers
53 views

Expectations and variance with rolling a dice 10 times

Let's say you roll a fair dice 10 times and X is the number of sides that never show up. (i.e. Roll 1 - 10 = 1424145221, X = 2 because 3 and 6 never show up) Values of $N=0,1,2,3,4,5.\\ P(N=6) = 0$ ...
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2answers
42 views

How does this summation hold?

How does $$ \sum_{r=0}^{m+n} \left( \sum_{k=0}^{r} \binom{n}{k} \binom{m}{r-k} \right) \ x^{r} = \sum_{r=0}^{m+n} \sum_{k=0}^{m+n} \binom{n}{k} \binom{m}{r} \ x^{r+k} $$ hold? RobJohn helped me, ...
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2answers
97 views

Value of the given limit

I need to calculate the value of : $$\lim_{n\to \infty}\frac{1}{n}\sum_{r=1}^{2n}{\frac{r}{\sqrt{n^2+r^2}}}$$ I had been trying to use Cesàro summation but somehow, I might be messing up. The ...
4
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2answers
99 views

Evaluating $\sum_{n=1}^{99}\sin(n)$ [duplicate]

I'm looking for a trick, or a quick way to evaluate the sum $\displaystyle{\sum_{n=1}^{99}\sin(n)}$. I was thinking of applying a sum to product formula, but that doesn't seem to help the situation. ...
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1answer
37 views

Calculating derivation of logarithm of summation of products

I am trying to grasp the idea discussed in this paper. In the second section of this paper it calculates the derivative of (1) which results in equation (2). I cannot figure out how the derivative of ...
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1answer
36 views

Binomial summation without coefficient

I have the following summation: $$\sum_{i=0}^na^{n-i}b^i$$ I recognise that if the binomial coefficient was present, it would represent the expansion of $(a+b)^n$. However, since that coefficient ...
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0answers
42 views

Gosper summable

I'd like to know why the following is NOT gosper summable: $$\sum_{k\in \Bbb{Z}} \frac{p(k)}{\prod_{j=0}^{m-1}(k+a+j)}$$ where $m>0, m\in\Bbb{Z}$ and $p(k)$ is a polynomial of degree $k=m-1$.
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4answers
460 views

Infinite sum of logarithms

Is there any closed form for this expression $$ \sum_{n=0}^\infty\ln(n+x) $$
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2answers
156 views

Approximating $\sum_{k=1}^{\infty}\dfrac{\sin(k^{1/n})}{k}\text{ for }n\in\mathbb{N}$

Again, inspired by this question, and the great answers I received here, I am curious as to why these infinite sums can be modelled with smooth functions. It appears that ...
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0answers
56 views

Question about factorial function [duplicate]

Show that $$n!=1+\left(1−{1 \over 1!}\right)n+\left(1−{1 \over 1!}+ {1 \over 2!}\right)n(n−1)+\cdots$$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove ...
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3answers
33 views

Generalize the number of total overall hops possible in a line network

Suppose I have a network of computers arranged in a line, like in the image I made below. I want to know the total number of hops possible. For example, $A$ gets to $B$ in $1$ hop, to $C$ in $2$ and ...
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7answers
341 views

Why does $\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}={\dfrac{\pi-1}{2}}$?

Inspired by this question (and far more straightforward, I am guessing), Mathematica tells us that $$\sum_{k=1}^{\infty}\dfrac{{\sin(k)}}{k}$$ converges to $\dfrac{\pi-1}{2}$. Presumably, this can ...
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2answers
45 views

Sum a series of series where each value increments by one

Can anyone suggest an elegant way to sum a series of numbers like this: (1, 2, 3, 4) (2, 3, 4, 5) (3, 4, 5, 6) (4, 5, 6, 7) That is for $n$ sets ...
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2answers
65 views

Simple factorial function question

Show that $n!=1+(1-1/1!)n+(1-1/1!+1/2!)n(n-1)+....$ I can't figure out how this can be solved . I tried to use binomial theorem but I couldn't prove it . Any help will be greatly appreciated.
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1answer
85 views

A frightening sum [duplicate]

Let $x_1,\ldots,x_r,y_1,\ldots,y_p,z_0,\ldots,z_r,t_0,\ldots,t_p$ be complex numbers. Let $A$ be the ring generated by these numbers. Prove the following holds in $\mathbb C(A)$. ...
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0answers
17 views

Converting proofs about finite or 'nice' sums to proofs over less nice sets. (Or: Converting marginal probability proofs to arbitrary event spaces)

I've been working on some basic probability problems. Two results that can be proved for finite or 'nice' (i.e. convergence works out nicely) event spaces by summing over one or more random variables ...
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4answers
246 views

Show $\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$ diverges

Show$$\sum_{k=1}^{\infty}\left(\frac{1+\sin(k)}{2}\right)^k$$diverges. Just going down the list, the following tests don't work (or I failed at using them correctly) because: $\lim ...
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2answers
34 views

Proving a trigonometric identity by infinite series

We know that: $$\cos(x) = \sum_{k=0}^\infty \dfrac{(-1)^kx^{2k}}{(2k)!}.$$ And $$\sin(x) = \sum_{k=0}^\infty \dfrac{(-1)^kx^{2k+1}}{(2k+1)!}.$$ Using this prove that $$\cos(x) \cdot \sin(x) = ...
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4answers
93 views

Prove the sum $\sum_{n=1}^\infty \frac{\arctan{n}}{n}$ diverges.

I must prove, that sum diverges, but... $$\sum_{n=1}^\infty \frac{\arctan{n}}{n}$$ $$\lim_{n \to \infty} \frac{\arctan{n}}{n} = \frac{\pi/2}{\infty} = 0$$ $$\lim_{n \to \infty} \frac{ ...
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1answer
25 views

How find this sum $\sum_{j=0}^{\infty}\binom{m+2j}{m}t^{2j},0<t<1$

Let $m$ is give postive integer numbers, Find the sum $$\sum_{j=0}^{\infty}\binom{m+2j}{m}t^{2j},0<t<1$$ if this not have closed form,and can you use Special function ?
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1answer
42 views

Prove that $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$

This is part of a proof itself. $\sum_{k=-\infty}^\infty e^{-j2\pi f k T}=\sum_{k=-\infty}^\infty\delta(f-\frac{k}{T})$ $\delta$ is Dirac function. It's been a while I am thinking about this part ...
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1answer
54 views

Find the set of convergence $\sum_{n=1}^{\infty} \frac{1+x^n}{1-x^n}$

How the interval [a, b]: $x \in [a,b]$ can be found for the next sum? $$\sum_{n=1}^{\infty} \frac{1+x^n}{1-x^n}$$ The sence to check the next limit $$\lim_{n \to \infty} \frac{1+x^n}{1-x^n} = ...
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1answer
27 views

$\sum^{\infty}_{n=1} \log(\frac{n+a+b}{n+a} \times \frac{n+b}{n+b+1})$

Prove $$\sum^{\infty}_{n=1} \log(\frac{n+a+1}{n+a} \times \frac{n+b}{n+b+1})=\log\frac{1+b}{1+a}$$ Hints/Answers are appreciated
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5answers
126 views

Find a closed form for $\sum_{k=0}^{n} k^3$ [duplicate]

Find a closed form for $\sum_{k=0}^{n} k^3$. I would appreciate ideas for approaching questions like this in general as well. Thanks.
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2answers
26 views

How should I solve this summation problem?

Lets say that we have these $x$ and $y$ coordinates $x=1,2,3,4,5$ and $y=6,7,8,9,10$ and where $n=5$. How would I use these $x$ coordinates with the first summation? Now, I know that learning is ...
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1answer
60 views

$\sum\limits_{i,j\in[|1,n|]}\frac{\max\left(i,j\right)}{\min\left(i,j\right)}$

I'm trying to do this sum $\sum\limits_{i,j\in[|1,n|]}\frac{\max\left(i,j\right)}{\min\left(i,j\right)}$ What I have tried : $\sum\limits_{i,j\in[|1,n|]}\frac{\max ...
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2answers
43 views

double summation problem $\sum^5_{i=1}i \times \sum^5_{j=1}j =…$ please check

(I) $\sum^5_{i=1}i \times \sum^5_{j=1}j = 1 \times (1) +1 \times (2) + \cdots +1\times (5) +2\times (1)+2\times (2) +\cdots + 2\times (5) + 3\times (1) + 3\times (2) + \cdots +3\times (5) + 4\times ...
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0answers
39 views

Prove that maximum 9 trailing zeroes in this summation

I am trying to prove that there are a maximum of 9 trailing $0$'s at the end of this summation: $$\sum_{k=1}^{k=m} k^n$$ for $1\le n\le 1000000$ and $m\le 100$. Any help on how to approach?
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3answers
24 views

Positive positive negative negative Series

What is the simplest series that alternates in order $+,+,-,-,+,+,-,- \dots$ Specifically I want to make a Riemann sum for something, but it has this reoccurent pattern I haven't previously ...
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1answer
118 views

$\sum\limits_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$ isn't divisible by 5

I have no idea Prove that for any $n$ natural number this sum $$\sum\limits_{k=0}^{n}\binom{2n+1}{2k+1}2^{3k}$$ isn't divisible by $5$. $\begin{array}{l} \left( {1 + x} \right)^{2n + 1} - ...
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5answers
110 views

Finding the minimum value of a sum [closed]

Let $x,y,z$ be real numbers . Find the real number $a$ so that $S$ has a minimum value , where $$S=|x-a|+|y-a|+|z-a| .$$
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0answers
31 views

Finite geometric sequence with a ratio greater than 1

I am trying to solve the recurrence relation: $ t(n)= 3T(n/2)+Cn$ (if $n>2$) $t(2)=C$ (otherwise) I know that there is the Master Theorem but I am trying to use the tree method. This ...
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1answer
67 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 ...
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1answer
77 views

Asymptotic of a sum evaluation as $ x \to \infty $

Let be the sum $$ \sum_{n\le x}[x/n]=g(x) $$ where $ [x] $ means floor function. My best try for asymptotic is $ g(x) \sim x\log (x)+\gamma x +1$ where I have used the asymptotic $ [x] \sim x $ ...
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1answer
48 views

Proof for $\sum_{x=1}^{n-1}\lfloor \dfrac{mx}{n}\rfloor=\dfrac{(n-1)(m-1)}{2}$ where $(m,n)=1$

This identity might be well-known, but I could find the proof neither by myself not by searching it in Internet. Could you describe an outline of solution?
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1answer
34 views

Summation formula in dimension 2

One of the most common tools in analytic number theory is the summation by parts, my question is what is the similar formula when we are, for example, in dimension two and we have the sum $$ ...
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0answers
34 views

What are the advantages/disadvantages of integration vs. summation?

If we are given a function, $f(x)$, we can either integrate it or sum it. I'm wondering what integration can do with $f(x)$ that summation can't, and what summation can do that integration can't. ...
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2answers
39 views

How to evaluate the following sum? $\sum_{i = 1}^n \left\lfloor \frac{3n-i}{2}\right\rfloor.$

What is the value of the following sum? $$\sum_{i = 1}^n \left\lfloor \dfrac{3n-i}{2}\right\rfloor.$$ Especially how to handle the sums with floors? This sum appeared while solving this problem. My ...
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3answers
61 views

How do I find the sum of the series?

$$\sum_{k=1}^{7}40 \left( \frac{1}{2}\right)^{k-1} = \frac{635}{8}$$ The image of the orginial eqn is on the link above and so is the answer, but I need help in how to solve it. when I did solve it I ...
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1answer
38 views

Can we possibly exchange summation and integration with negative values?

This is an attempt to go further than this answer. Essentially, we have either a summation of an integral: $$\sum_x{ \left( \int{ f(x)dx } \right) } \tag{1}$$ ...or an integral of a summation: ...
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1answer
39 views

Is it possible to get a formula for this summation

The binomial sum $$s_n=\binom{n}{0}+\binom{n+1}{1}+\binom{n+2}{2}+\cdots+\binom{2n}{n}$$ I tried solving through recurrence, but unable to find one.
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2answers
182 views

How to prove this series $\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$ diverges

Question: Assume that $a_{n}>0,n\in N^{+}$, and that $$\sum_{n=1}^{\infty}a_{n}$$ is convergent. Show that $$\sum_{n=1}^{\infty}\dfrac{a_{n}}{(n+1)a_{n+1}}$$ is divergent? My idea: since ...
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1answer
37 views

Complex summation simplification

What I'm getting is $$\frac{( \sin (N+1)x - 2^N \sin x)}{(2^N(\sin x - 2))}$$ How do I simplify to the form they have given , please help. I hope it's clear because I don't know Ajax still ...
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1answer
63 views

When can we use substitution for both integrals and summations?

This question is partially inspired by Qiaochu Yuan's answer to "Will moving differentiation from inside, to outside an integral, change the result?". Essentially, I would like to know, if we have: ...
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1answer
33 views

Complex number summation.

$$ \sum_{n=1}^N\cos(2n-1)\theta=\dfrac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0.$ Deduce that $$ \sum_{n=1}^N (2n-1)\sin\left[\dfrac {(2n-1)\pi}N\right]=-N\operatorname{cosec}\dfrac\pi ...
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3answers
56 views

Find value of x in the given expression [closed]

Find the value of x in the following expression 2^2 * 2^6 * 2^10 * ..... *2^x = (0.125)^-24
1
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1answer
49 views

Summation of cos (2n-1) theta

By considering $\sum\limits_{n=1}^N z^{2n-1}$, where $z=e^{i\theta},$ show that $$ \sum\limits_{n=1}^N \cos{(2n-1)} \theta = \frac{\sin(2N\theta)}{2\sin\theta}, $$ where $\sin\theta\neq0$ I ...
1
vote
3answers
217 views

Sum of digits of number from 1 to n

Is there any general formula for calculating the sum of digits of number from 1 to n? n < 10^9