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

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0
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2answers
28 views

Evaluate the sum below

Evaluate the following sum $$1*1!+2*2!+3*3!+....+1000*1000!$$ any help guys?
0
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0answers
20 views

Why is $\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$

Let $T$ be a non-negative random variable. Why is it true that $$\sum_{k=1}^{\infty}\mathbb{E}[\mathbb{1}(T=k)]=\sum_{k=0}^{\infty} k \mathbb{P}[T=k]$$ According to me it would make sense that ...
2
votes
2answers
36 views

Why does this sum equal zero?

Le}t $\gamma$ be a piece-wise, smooth, closed curve. Let $[t_{j+1}, t_{j}]$ be an interval on the curve. Prove, $$\int_{\gamma} z^m dz=0$$ In the proof it states $$\int_{t_{j}}^{t_{j+1}} ...
0
votes
0answers
22 views

removing a factor when the summation equals zero

http://www2.warwick.ac.uk/fac/soc/economics/staff/vetroeger/teaching/po906_week567.pdf On page 9 of the above link. I don't understand the step from row 5 to row 6. After the x_i is pulled out as a ...
1
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2answers
32 views

Find $S=\sum\sum\sum x_{i}x_{j}x_{k}$ where $x_{i}=-x_{n-i+1}$ for $1\leq i\leq n$

Suppose that $x_{1},x_{2}.....x_{n},(n>2)$ are real numbers such that $x_{i}=-x_{n-i+1}$ for $1\leq i\leq n$. Consider the sum $S=\sum\sum\sum x_{i}x_{j}x_{k}$, where the summation are taken over ...
2
votes
2answers
377 views

Changing from Positive to Negative

I may mess up a little bit...Sorry for that! When we want a summation to go negative in case of odd numbers and positive otherwise , we use: $$\sum\limits_{i=1}^{12} \color{red}{{(-1)}^i} 2x^3 $$ ...
0
votes
3answers
66 views

Prove sum of $\sin$ of angles is greater than $\sin$ of sum of angles

It seems that $\displaystyle \sum_{x_i \in X} \sin\left(x_i\right) \geq \sin\left(\sum_{x_i \in X} x_i\right)$ where $X$ is a set of angles where $\displaystyle \sum_{x_i \in X} x_i \leq \pi$ radians ...
0
votes
4answers
57 views

Can you prove the simplification of this sum?

I'm still learning Calculus (in parallel) and I'm stuck on this sum simplification. It is the 2nd part of the Tail to expectation formula from statistics #1. From here : $$ \sum_{k=a+1}^{b} \frac{ ...
0
votes
4answers
138 views

How to derive $\sum_{n=0}^\infty 1 = -\frac{1}{2}$ without zeta regularization

On Wikipedia we find $\displaystyle \bbox[5px,border:1px solid #F5A029]{1 + 1 + 1+\dots =\sum_{n=0}^\infty 1 = -\frac{1}{2}}$ using (the rather complicated) zeta-function regularization. I asking for ...
1
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2answers
82 views

What is the sum of all real numbers from $0$ to $1$? [on hold]

I wanted to know the approximate sum of real numbers from 0 to 1. Please tell me how we can find it.
2
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0answers
18 views

Alternating sum of subfactorials: Is there a closed form for this: $\displaystyle \sum_{i=0}^{m-2}(-1)^i\left[\frac{(m-i)!}{e}\right]$?

The problem was to find the number of ways in which $n$ objects in circular arrangement can be placed so that each one has a new object in front of it (assuming a particular, initial arrangement). ...
1
vote
1answer
37 views

Help with a summation+inequality problem.

I need help in solving for all possible x values for the below inequality: (Note: $x \in N)$ $$\sum^x_{k=1}\frac{k^2+k+1}{k(k+1)(k+1)!} \leq \frac{599}{600}$$ I think the series is telescopic; I'm ...
-1
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6answers
118 views

What is the mathematical symbol for the sum of numbers

For example, when $n=5$, what is the symbol for $5+4+3+2+1$?
1
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0answers
38 views

how to sum the floors of ratios n/k when prime factorization of n is known

According to @harald-hanche-olsen the sum of the floors of ratios of $n/k$ is approximately: $$n(\ln n-1-\ln2)<\sum_{k=2}^{n-1}\Bigl\lfloor \frac nk\Bigr\rfloor<n\ln n.$$ If the prime ...
1
vote
5answers
216 views

I need help with a Finite Series

Problem: Find the sum to $n$ terms of \begin{eqnarray*} \frac{1}{1\cdot 2\cdot 3} + \frac{3}{2\cdot 3\cdot 4} + \frac{5}{3\cdot 4\cdot 5} + \frac{7}{4\cdot 5\cdot 6}+\cdots \\ \end{eqnarray*} ...
5
votes
2answers
122 views

Limit involving binomial coefficient

I was trying to find the below limit. The sum can be written in a hypergeometric function but it doesn't seem to help me to find the limit. Any help will be appreciated. $$ \lim_{n \rightarrow ...
2
votes
3answers
86 views

Binomial Relation: $\sum_{n=0}^N\binom{N}{n}(-1)^{N-n}n^k=0$

I need this as lemma. Regard the sums: $$S_k:=\sum_{n=0}^N\binom{N}{n}(-1)^{N-n}n^k\quad(k\in\mathbb{N}_0)$$ Then it holds: $$S_k\stackrel{k<N}{=}0\quad S_k\stackrel{k=N}{=}N!$$ How can I check ...
2
votes
1answer
41 views

Minkowski's inequality

Minkowski's inequality for sums states $$\left(\sum_{j=0}^\infty |a_j+b_j|^2 \right)^{1/2} \le \left(\sum_{j=0}^\infty |a_j|^2 \right)^{1/2}+\left(\sum_{j=0}^\infty |b_j|^2 \right)^{1/2} $$ for ...
0
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1answer
25 views

Proving a function is continuous and periodic

Suppose we are given a function $$g\left ( x \right )= \sum_{n=1}^{\infty}\frac{\sin \left ( nx \right )}{10^{n}\sin \left ( x \right )},x\neq k\pi , k\in\mathbb{Z}$$ and $$g\left ( k\pi \right ...
3
votes
2answers
56 views

The sum of squares of the first $n$ natural numbers.

My basic question is this: how to find the sum of squares of the first $n$ natural numbers? My thoughts have led me to an interesting theorem: Faulhaber's formula. It is known that ...
4
votes
3answers
63 views

Help with a Series (Edited)

The original problem was: $$\sum_{k=0}^\infty\dfrac{k}{6k^3+13k^2+9k+2}$$ Using Partial Fractions, I resolved this into ...
3
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2answers
43 views

When are both $\sum_{n=0}^\infty \log(a_n)$ and $\sum_{n=0}^\infty a_n$ convergent?

I'm new to this site. Can someone give me some examples of when both: $$\sum_{n=0}^\infty \log(a_n)\qquad \text{ and }\qquad \sum_{n=0}^\infty a_n$$ are convergent?
0
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0answers
22 views

The Summation of a Summation in Mathematica [on hold]

I am trying to input $\sum_{a_1=1}^{n-1}$$\sum_{a_2=1}^{{a_1}-1}$$\sum_{a_3=1}^{{a_2}-1}$$\sum_{a_4=1}^{{a_3}-1}$$...f({{a_1},{a_2},{a_3},{a_4}...})$ into mathematica. The number of sums should be a ...
7
votes
3answers
311 views

Solve infinite series equation with logarithmic terms.

Solve logarithmic equation: $$\frac{\log x^2}{\log^{2}x}+\frac{\log x^3}{\log^{3}x}+\cdots+\frac{\log x^k}{\log^{k}x}+\cdots=8$$ here $\log$ is assumed to have base $10$. So far I managed to rewrite ...
0
votes
1answer
24 views

Limit of summation as n goes to infinity

I am trying to solve the following: Let $q>1$ and $n \in N$. Evaluate $\lim_{n \rightarrow +\infty} \sum_{k=1} ^n \frac{k^{q-1}}{n^q + k^q}$. I understand that I need to first get the summation ...
8
votes
1answer
123 views
+50

How do I prove this combinatorial identity using inclusion and exclusion principle?

$$\binom{n}{m}-\binom{n}{m+1}+\binom{n}{m+2}-\cdots+(-1)^{n-m}\binom{n}{n}=\binom{n-1}{m-1}$$ Note that we can show this with out using inclusion and exclusion principle by using Pascal's Identity ...
2
votes
1answer
61 views

Coefficients of a generating function

I need a bit of help. I was solving the form of the coefficients of the generating function $\sum_{n}n^m r^n$. Then I started building the indefinite sum $\sum n^m r^n \delta n$ trough recursive ...
5
votes
3answers
73 views

Infinite sum of alternating telescoping series

I am struggling to find the sum of the following series: $$\sum_{n=1} ^{\infty} \frac{(-1)^n}{(n+1)(n+3)(n+5)}.$$ It seems as though it should be a straightforward telescoping series. I attempted to ...
2
votes
3answers
472 views

Change from product to sum

We know that : $$a \times b = \underbrace{a + a + a + ... + a}_{\text{b times}}$$ That's how we convert from a product to a sum. So what happens if we go a little further? That is : ...
3
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1answer
45 views

closed form for some binomial sum

I am trying to derive a closed form for the generating function of $a_n(x)=\sum_{k=0}^n \binom{n+k}{n}x^k, x>0, n\in\mathbb{N},$ i.e. for $G(z)=\sum_{n=0}^\infty a_n(x)z^n$. The only method I ...
1
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0answers
11 views

Simple question about derivative and summation

I am reading a book and I have a simple question. There is this summation: $$ A = β\sum_\textbf{x} ||\textbf{x}||^2 r(\textbf{x})$$ after this, taking the partial derivative: $$ \frac{\partial A ...
0
votes
2answers
37 views

Summation of powers

I have come across the following in my textbook: $$\sum_{i=0}^{20} 5^i = \frac{5^{21}-1} {4} $$ There is no explanation of how this result was achieved. Could anyone help walk-through how this would ...
2
votes
1answer
66 views

Summation by Parts to Evaluate $\sum_{k=1}^{\infty}(2k+1)x^{2k}$

I need to evaluate $\sum_{k=1}^{\infty}(2k+1)x^{2k}$ using the Summation by Parts (SBP) method. It is given that $0 < |x| < 1$. The notation our class uses for SBP is as follows: $$ \sum_{i} ...
0
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1answer
51 views

How simplify this particular sum?

Can we simplify the following sum? $$\sum_{i=1}^n \binom{n}{i} {(-1)^{i+1}\over 1-2^{-i}}$$ Thank you.
1
vote
1answer
98 views

Reversing the Order of Integration and Summation

I am trying to understand when we can interchange the order of Integration and Summation. I am increasingly encountering Integrals; some of which are being solved by interchanging the order of ...
0
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0answers
8 views

Force directed graph on a wrapped plane

I'm writing a force-directed graph where the plane is wrapped. Physics-wise this should cause the resulting forces to be an infinite sum based on each original node's distances to every recurrence of ...
-2
votes
3answers
63 views

Solving sum of $(-1)^n (1/2)^n$ [closed]

How to solve the following sum? $$\sum_{n=0}^k (-1)^n (1/2)^n$$
2
votes
1answer
35 views

Expanding summation $\sum_{i=1}^{k+1}i(i!)$

Expand the summation: $\sum_{i=1}^{k+1}i(i!)$ My solution is: $\sum_{i=1}^{k}i(i!)+k(k+1)$ But I think it is wrong. Please help. Thanks
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0answers
17 views

Is a norm of a vector a constant to a summation?

I have a simple question: if we have $$\sum_{\textbf{x}}\|\textbf{x}\|^2\, p(\textbf{x}),$$ can I treat the norm as a constant and remove it from the summation?
0
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1answer
23 views

Taking it a step further with a sum

So I was watching an "old" video from numberphile about the three square problem. https://youtu.be/m5evLoL0xwg Here is also an image: ...
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0answers
92 views

A closed form for $\int x^nf(x)\mathrm{d}x$

When trying to find a closed form for the expression $$\int x^nf(x)\mathrm{d}x$$ in terms of integrals of $f(x)$ I found that $$\int xf(x)\mathrm{d}x=x\int f(x)\mathrm{d}x-\iint ...
1
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1answer
54 views

Problem with this challenging summation

I'm having some trouble finding the summation of this series. I tried all I could, but in the end the denominator is creating problem. $$ \sum_{r=0}^{n} (-1)^r ...
1
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0answers
54 views

Solving summation; ( double sum).

I found the expression to the sum of powers long ago and ofcours I think it is true but i don't know for sure, the problem is, it's little though for me to test and try it out. Also i'd like to know ...
4
votes
1answer
89 views

Fnd the sum of first 99 terms of a sequence

Find the sum of first $99$ terms of a sequence, where $$T_{n}=\frac{1}{5^{2n-100}+1}$$ I need some hints on how to approach, I am unable to simplify it. Thanks.
1
vote
1answer
25 views

Eliminating a summation

I need to approach the new position $(x_t,y_t)$ at moment $t$ of a moving object at $(x_0,y_0)$ given its horizontal velocity $vx_0$, its vertical velocity $vy_0$ and some constant resistance $r$ that ...
0
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0answers
28 views

Checking if the sum of a set of functions can ever be zero

Given a set of elementary functions $f_1(x), f_2(x), f_3(x), ..., f_n(x)$ that are known to either be zero at some point, or always nonzero, it is trivial to check if the product of all these ...
0
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1answer
46 views

Combinatorial proof for $\sum_{k = 0}^n \binom {r+k} k = \binom {r + n + 1} n$ [duplicate]

I'm trying to figure out a combinatorial proof for: $$\displaystyle \sum_{k \mathop = 0}^n \binom {r+k} k = \binom {r + n + 1} n$$ I've tried the committee counting thing, but that didn't work.
1
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0answers
20 views

Fourier cosine transformation

Good day! I'm studying right now some transformation and I encountered the following equation: $$(2\pi)^{-n/2} \int_{-\infty}^\infty\cdots\int_{-\infty}^\infty \exp\left(-\frac{1}{2} ...
0
votes
2answers
29 views

Is there a shortcut to summing fractional powers?

I'm trying to solve a problem where, incrementally, each step sums a particular value. The value is plagued with an ugly fractional power. Is there a shortcut to something like $$\sum_{i=0}^n ...
0
votes
1answer
19 views

Special Form of Combination - Formula Verification

Giving abc how many combination that involves a ? answer is a ab ac abc equal to 4 I came up with the following formula but I would like to know of its correctness plus if there is simpler form ...