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
54 views

Find the result of $ \sum_{x=1}^\infty \frac{1}{x} \log \frac{kx}{\lfloor kx \rfloor}_o$

I would like to calculate the sum $$\displaystyle \sum_{x=1}^\infty \frac{1}{x} \log \frac{kx}{\lfloor kx \rfloor}_o$$ where $k=\sqrt{2}+1$, $x$ is an odd integer and $\lfloor z \rfloor_o$ indicates ...
0
votes
1answer
76 views

Prove with combinatorial arguments this equation [duplicate]

Prove with combinatorial arguments, that, $\forall n \in \mathbb{N}$. $$\sum_{k=0}^n (-1)^k {n \choose k} =0$$
2
votes
0answers
30 views

Necessary condition for symmetric sums

It is easy to see that if a function $f(x_1,x_2,x_3)$ can be written in the form: $$ f(x_1,x_2,x_3) = g(x_1,x_2) - g(x_1,x_3) + g(x_2,x_3) $$ for some function $g$, then we have: $$ f(x_1,x_2,x_3) - ...
0
votes
0answers
29 views

How to construct a function to map coefficients?

Surely this question is known by many people but I lack of enough maths knowledge so I prefer ask here. I have a triangular matrix that represent coefficients, all of them are rational numbers ...
0
votes
1answer
44 views

Sum of the series of $\displaystyle \sum_{i=0}^ni^2 2^{n-1}$

Question: Evaluate $$ \sum_{i=0}^n i^2 2^{n-i}$$ In the previous questions on my question paper I have got: $$\sum_{i=1}^n a^i = \frac{a(1-a^n)}{1-a}$$ and ...
1
vote
0answers
63 views

Double summation of a geometric series

I am interested in the following sum for a given value of n: $ \sum\limits_{x=1}^{n} \sum\limits_{y=1}^{n}x^y$ I can simplify this to $ \sum\limits_{x=1}^{n} \frac{x^{n+1} - x}{x - 1}$ From here ...
1
vote
1answer
53 views

$\sum_{k+M = 0}^n {n \choose k} {n-k \choose m} = 3^n$ help with combinatorial reasoning

$\sum_{k+M = 0}^n {n \choose k} {n-k \choose m} = 3^n $ I have worked the cases for $n=2$, $n=3$, and $n=4$ by hand and it appears to be true. $n=2$: $${2\choose 0}{2\choose 0} + {2\choose ...
0
votes
2answers
57 views

Closed form for $1/(n (n + 1))$

What would the closed form be for $$ g(n) = \frac{1}{1 \cdot 2} + \frac{1}{2 \cdot 3} + ... + \frac{1}{n \cdot (n+1)}\;\;? $$ An example would be $g(3) = \frac{1}{2}+\frac{1}{6}+\frac{1}{12} = ...
2
votes
2answers
55 views

Infinite sum of ratio of factorials

Mathematica tells me that for $r, p$ positive integers with $\mathcal{r} < p$ we have $$\sum_{i=p+1}^\infty \frac{(i-r)!}{(i+r)!} = \frac{(p+1-r)!}{(2r-1)(p+r)!}. $$ Can anyone point me in the ...
0
votes
2answers
156 views

Reversing the order of summation in $\sum_{i=1}^{n} i$

Here is a summation that is supposed to be solved: $$S_n = \sum_{i=1}^{n} i$$ The author says it can be solved by substituting with $i = n-j$: $$\sum_{i=1}^{n} i = \sum_{n-j=1}^{n} (n - j)$$ The ...
1
vote
3answers
72 views

Summation of $2^n$ where $n$ is even or odd

I need the summation formula for the following: $2^0+2^1+2^3+2^5+\cdots+2^n$, $n$ is even i.e. $2^0+2^2+2^4+2^6+\cdots+2^n$, $n$ is odd I am aware that the formula of $2^0+2^1+2^2+2^3+\cdots+2^n ...
1
vote
1answer
70 views

Summation of: $\sum_{1}^{\infty}\left(\frac{2}{3}\right)^x$ [duplicate]

This is a subsection in my statistics homework. It goes back to calculus II and summations, and it's been a long time since I've studied it so I'm rusty. I'm looking to solve the summation of ...
1
vote
2answers
48 views

Add integers to a set number of lists, so that the sum of each completed list is as closely matching to the other lists as possible?

I am trying to figure out how to solve this problem in computer science. I won't go into the programming side of things, but basically what I need is this: I have a list of integers ranging from ...
2
votes
0answers
48 views

Closed form for the sum $\sum_{a=1}^{b} a^3\cdot (b \bmod a)$

How can we simplify $\sum_{a=1}^{b} a^3\cdot (b \mod a)$? For $a \ge \frac{b+1}{2} $ to $a = b$ it reduces to $$\sum_{a\ge \frac{b+1}{2}}^{b}a^3\cdot (b-a)=b\cdot\sum_{a\ge ...
3
votes
1answer
171 views

How to compute this sum?

I want to sum the following: $$f(n) = \sum_{i=1}^n (i^3 \cdot (n \mod i))$$ Since the sum can be huge I have to output the sum modulo some given number m. How can I approach this problem? Also, n ...
-2
votes
2answers
59 views

How to show that $ \sum_{d/n} \mu^{2}(d)/\phi(d) = n/\phi(n)$? [closed]

$\forall n, n\in\mathbb{N}$ $\frac{n}{\phi{(n)}} = \sum_{d/n} \frac{\mu^{2}(d)}{\phi(d)}$ Where $\mu$ is the Möbius function.
0
votes
4answers
48 views

Sum of squares of sines.

Any ideas on how to compute this sum? I'm sure there's a simple trick to it, but I just can't wrap my mind around it at the moment. Some insight would be tremendously appreciated, thanks! ...
-2
votes
1answer
14 views

Reducing sum with different coefficients

Is it possible to further reduce this sum? $5(x+y) + 4(z+w) + 3(l+m) + 2(n+o) + (p+q)$ Where $x, y, z, w, l, m, n, o, p, q$ are always different?
0
votes
2answers
27 views

Factorial Summation Problem [duplicate]

$$\sum_{j=0}^n j\cdot j!$$ I got $(n+1)!-1$ as the answer but I'm not sure if that's right or how I even got to that answer exactly. (my paper is a mess of random work and I can't make it out). Can ...
0
votes
2answers
50 views

Asymptotics of $\sum _{k=1}^n \sum _{j=1}^n \frac{j k}{j+k}$

I'm asked to find a simple asymptotical estimation of $\displaystyle \sum _{p=1}^n \sum _{q=1}^n \frac{p q}{p+q}$. I rewrote the sum as $\displaystyle \sum _{k=2}^{2 n}\sum_{p+q=k}\frac{pq}{p+q}= ...
2
votes
5answers
122 views

Closed form for $1 + 3 + 5 + \cdots +(2n-1)$ [duplicate]

What is the closed summation form for $1 + 3 + 5 + \cdots + (2n-1)$ ? I know that the closed form for $1 + 2 + 3+\cdots + n = n(n+1)/2$ and I tried plugging in $(2n-1)$ for $n$ in that expression, ...
1
vote
4answers
44 views

Why is $\sum\limits_{b=1}^{t-1} {t \choose b} 2^{t-b} = (3^t - 2^t - 1)$

Why is $$\sum\limits_{b=1}^{t-1} {t \choose b} 2^{t-b} = (3^t - 2^t - 1)$$ Thanks.
0
votes
0answers
31 views

Recurrence Relation Partitioning 8 into 4 parts

Where do I begin with writing down all the partitions of this equation? Let $P(r,n)$ denote the number of partitions of r into n parts. Use the recurrence relation $$P(r, n) = \sum_{i=1}^n ...
1
vote
2answers
70 views

Sum of a geometric series

What is the step of solving this problem? Evaluate: $$\sum_{i=1}^nia^i$$ For what I know, it's $$\sum_{i=1}^ni\times\sum_{i=1}^na^i$$ I know how to evaluate ...
4
votes
4answers
131 views

Value of $\psi\left(\frac{1}{2}\right)$

I apologise if this is a dumb question, but I have trouble deriving $\displaystyle\psi\left(\frac{1}{2}\right)=-\gamma-2\ln{2}$. I have tried the following. \begin{align} \psi\left(\frac{1}{2}\right) ...
3
votes
4answers
81 views

how to solve $\sum _{m=0}^{k-1}mC_{k-1}^{m}C_{N-k}^{m}$?

solving $$\sum _{m=0}^{k-1} mC_{k-1}^m C_{N-k}^m$$ the solution seems to be $$\frac {\left( N-2\right) !} {\left( k-2\right) !\left( N-k-1\right) !}$$ according to some clue from the other ...
3
votes
1answer
100 views

How do I separate the summand of a double summation?

For example, if I have: $$\sum_{i=1}^n \sum_{j=1}^n (x_i^2 - 2x_ix_j + x_j^2)$$ How would I separate these terms? Unfortunately, we've dived into proofs using summation without much background ...
5
votes
5answers
224 views

Need to prove inequality $\sum\limits_{k=0}^n \frac{1}{(n+k)} \ge \frac{2}{3}$

Prove that for $n \geq 1$: $$\sum\limits_{k=0}^n \frac{1}{(n+k)} \ge \frac{2}{3}$$ I have tried math induction but that didn't work. Although I'm pretty sure that the solving can be done by induction ...
1
vote
3answers
327 views

Using Binomial Theorem to prove the following [duplicate]

$$\large\sum_{j=0}^n (-1)^j {n\choose j}={n\choose 0}-{n\choose 1}+.....+\pm{n\choose n}=0 $$ I'm confused by the last part of the equation $\pm$. it seems imply that the sum would be equal to 0 no ...
0
votes
1answer
45 views

How to solve matrix eigenvalue equation which has a summation.

General problem: If I have some $n \times n$ matrices $\mathsf{M}^\tau$, and column vectors (with $n$ rows) $X^\tau$ is there some mathematical tricks I can do to solve the eigenvalue equation $ ...
0
votes
3answers
74 views

Is it correct that $ \sum_{i=m}^na=(nm+1)a$ [closed]

I study this in a book. Is it correct? Why? $$ \sum_{i=m}^na=(nm+1)a$$
1
vote
3answers
70 views

Find the sum $1+\cos (x)+\cos (2x)+\cos (3x)+…+\cos (n-1)x$ [duplicate]

By considering the geometric series $1+z+z^{2}+...+z^{n-1}$ where $z=\cos(\theta)+i\sin(\theta)$, show that $1+\cos(\theta)+\cos(2\theta)+\cos(3\theta)+...+\cos(n-1)\theta$ = ...
6
votes
0answers
130 views

Second derivative of Hypergeometric function

I'm looking for the following second derivative $$ \kappa_2 := \left . \frac{d^2}{d\lambda^2} \ln \left({_2F_1}\!\left(\tfrac{1}{2},\,- \lambda;\,1;\,\alpha\right)\right) \right \vert_{\lambda = ...
0
votes
1answer
60 views

Sum of Taylor Series

I have the converging series: $$ 1 - x + \frac{x^2}{2!} - \frac{x^3}{3!}+... $$ and I'm trying to find its sum when x = .9. I know this is the Taylor series for some function$f(x)$, and that I can ...
2
votes
2answers
69 views

Factorial as a sum. Insight appreciated

I recently posted an answer to a question about ways to express the factorial function as a sum. I posted the following formula, which I discovered several years ago and I haven't seen anywhere else: ...
0
votes
0answers
18 views

Sum over stochastic processes on the same set of categories

I have a stochastic process consisting of multiple (stochastic) steps, for which I want to know if I can substitute (or at least approximate) it by summing over the deterministic and stochastic parts ...
1
vote
4answers
31 views

Evaluating Summation of $5^{-n}$ from $n=4$ to infinity

The answer is $\frac1{500}$ but I don't understand why that is so. I am given the fact that the summation of $x^{n}$ from $n=0$ to infinity is $\frac1{1-x}$. So if that's the case then I have that ...
-1
votes
1answer
72 views

Prove ${-1 \choose n} = (-1)^n$ [closed]

For my Math Physics homework, I have to show that ${-1\choose n} = (-1)^n$. I read the textbook (no help) and this wasn't covered in lecture, I have no idea where to even begin. Please help.
1
vote
1answer
64 views

When can we exchange limits and summations?

I'm hoping to find a lot of information for the case when we can say: $$\lim_{a \to 1}{ \sum_{k=b}^c{f(a,k)} } = \sum_{k=b}^c{ \lim_{a \to 1}f(a,k) }$$ ...or, more generally, when $a$ approaches ...
1
vote
3answers
51 views

Summing $\sum_{n=0}^{\infty} n x^n$ [duplicate]

I am trying to figure out how to do the infinite summation: $$ \sum_{n=0}^{\infty} n x^n \qquad 0 \leq x < 1$$ The series converges so it seems to me that the limit must exist, but I'm having ...
2
votes
1answer
102 views

Challenge: How to prove this reduction identity for factorials of even numbers?

Some time ago, as a by-product of a proof, I came across an odd (at least to me) identity for reducing the factorial of an even number into a sum: $$(2n)!=\sum_{k=0}^{\lfloor \frac{n}2 \rfloor} ...
6
votes
1answer
141 views

Prove |cos(x−1)|+|cos(x)|+|cos(x+1)|≥3/2

I'm working on an induction proof, but I keep coming up against a brick wall. While working through the induction proof process I keep ending up with $$|\cos(m)|\ge\frac12$$ ,but clearly this isn't ...
4
votes
2answers
54 views

Nested Sum Encountered in Maclaurin Expansion of $e^{-x^2}$

In the course of working out the Maclaurin expansions of $e^{-x^2}$ and $cos(x^2)$, I ran into the following nested sum: $$ \underbrace{ \sum_{a=0}^1 \left( a \sum_{b=0}^{a+1} b \left( ...
5
votes
3answers
132 views

Use the binomial theorem to show that for any positive integer $n$, $\displaystyle\sum_{i=0}^{n} {n \choose i} = 2^n$.

Can somebody check to see if this is good enough just to show? It's very simple but the question doesn't say prove or anything like that. So the binomial theorem states that ...
7
votes
1answer
199 views

Sum $\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$

I would like to seek your assistance in computing the sum $$\sum^\infty_{n=1}\frac{(-1)^nH_n}{(2n+1)^2}$$ I am stumped by this sum. I have tried summing the residues of $\displaystyle ...
5
votes
1answer
97 views

Number of pairs $(i, j)$ where $1\leq i < j \leq N$ such that $i|j$

This is a programming question so I tried like this: for (int i=1; N/i != 1; i++) sum += N/i-1; I am getting the correct answer. But one solution I saw by ...
3
votes
3answers
133 views

Evaluate the sum

I need to evaluate the following sum, which depends on $n \in \mathbb N$ (call it $S(n)$ if you will) $$ \sum_{i=0}^{n} (-1)^{n-i} \binom{n}{i} f(i)$$ where $f$ defined over $\mathbb N$ is ...
1
vote
2answers
68 views

Using induction to prove that $\sum_{r=1}^n r\cdot r! =(n+1)! -1$

Use induction to prove that $\displaystyle\sum_{r=1}^n r\cdot r! =(n+1)! -1$ I first showed that the formula holds true for $n=1$. Then I put n as $k$ and got an expression for the sum in ...
1
vote
2answers
69 views

Question of the value of the following double sum

I hope to get the exact value of the following double series: $$ \sum_{k \geq 0} \sum_{n \geq 0} \binom{2k}{ k} \frac{(-1)^n}{n! (2k+2n+1) 2^{4k+2n+1}}. $$ I am not sure it is possible or not. I ...
3
votes
1answer
68 views

$n$th erivative of $f(x)^{n+1}$

Hello, I'm trying to find the $n$th derivative of a function, where $n\in\mathbb N$ $$\frac{d^{n-1}}{dx^{n-1}} \!\!\left[f(x)^n\right]$$ I'm looking for some kind of sum or product (or nesting ...