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

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6
votes
2answers
47 views

$\sum_{k=1}^n \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$

(Own) Let $n,m$ be positive integers such that $m>n$. Prove that $$\sum_{k=1}^n \sum_{a_1+a_2 + \cdots +a_k=n} \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$$ ...
1
vote
2answers
62 views

Sum of super exponentiation

$f(x,n)=x^{2^{1}}+x^{2^{2}}+x^{2^{3}}+...+x^{2^{n}}$ Example: $f(2,10)$ mod $1000000007$ = $180974681$ Calculate $\sum_{x=2}^{10^{7}} f(x,10^{18})$ mod $1000000007$. We know that $a^{b^{c}}$ mod ...
0
votes
0answers
26 views

Why does the Harmonic series diverge? [duplicate]

I am aware of Oresme's proof of its divergence(http://mathworld.wolfram.com/HarmonicSeries.html), but this proof could be applied to the sum of all natural numbers and it would still be valid. Yet, ...
0
votes
1answer
26 views

What is the proof for this sum of sum generalized harmonic number?

I believe this sum: $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}$$ to be equal to $$\frac 12((H_k^{s})^2-H_k^{2s})$$ where $H_k^{s}$ is the generalized harmonic number. I only discovered this by ...
2
votes
2answers
50 views

Forming natural numbers with positive consecutive integers

I'm trying to prove that any natural number N can be formed by adding at least two positive consecutive integers except for powers of 2. For example, using $\,N = 3$, $N = 1 + 2$. When experimenting ...
4
votes
1answer
78 views

A Combinatorial Sum!

Is there a closed form formula for the following sum \begin{equation} F(x;n,m)=\sum_{k=0}^{\min\{n,m\}} {n \choose k}{m \choose k}k!\ x^{k}=n! \, m!\sum_{k=0}^{\min\{n,m\}}\frac{1}{k!(n-k)!(m-k)!} ...
0
votes
4answers
96 views

why $ \sum_{k=0}^{\infty} x^{2k} = \frac{1}{1-x^2}\\$

Why $$ \sum_{k=0}^{\infty} x^{2k} = \frac{1}{1-x^2}\\$$ I know that $$ \sum_{k=0}^{\infty} x^k = \frac{1}{1-x}\\$$ can I use the above to derive the first result?
0
votes
0answers
29 views

How to separate self-defining values from sigma?

$$\sum_{k=1}^{m} \sum_{j=1}^{n} a_kx_j^{b_k+b_i} = \sum_{j=1}^{n} y_jx_j^{b_i}$$ What I need to do is solve $a$ for every $i$ given ($i$ is between 1 and $m$), so their result won't be composed of ...
1
vote
3answers
44 views

Finding $\lim\limits_{n\to \infty}({1\over n+1}+{1\over n+2}+…+{1\over n+n})$ using integrals [duplicate]

Finding $\lim\limits_{n\to \infty}\left({1\over n+1}+{1\over n+2}+\dots+{1\over n+n}\right)$. I tried many things but it would work out. I am now studying calculus 2 (In my country the first calculus ...
1
vote
2answers
53 views

Studying this sum: $\sum_{k=0}^{\infty} (k+1)(x)^k$

$$\sum_{k=0}^{\infty} (k+1)(x)^k\Longrightarrow x=-\frac{9}{10} \Longrightarrow\sum_{k=0}^{\infty} (k+1)\left(-\frac{9}{10}\right)^k=\frac{100}{361}\approx 0.2777$$ $$\sum_{k=0}^{\infty} ...
0
votes
0answers
26 views

Infinite summation (sum of natural numbers) [duplicate]

I've proof proof that this is equal but how can I proof it, and is it even equal?! $$\lim_{s\rightarrow0}\left(\sum_{k=0}^{\infty} (k+1)k^{-s}(-1)^k\right)=\sum_{k=0}^{\infty} ...
0
votes
1answer
34 views

$f,g \in R(T)$ such that $\hat{f} \cdot n^{2/3} = \hat{g}$ prove that $f$'s Fourier series converges absolutely.

Can someone help me by checking my solution. Is there a shorter More elegant solution ?(i'm almost sure you can some how express $f$'s Fourier series using $|\hat{g}|^2$ + constant, i saw someone do ...
1
vote
0answers
32 views

How to manipulate the bound on the summation

$$ B_n^{f^2}(x) = \sum_{k=1}^n\sum_{j=0}^{n-k} 2^{k-j} {j+k \choose j} \frac{d^j}{df^j}[f^k] B_{n,j+k}^f(x) $$ I am looking to have the bounds switched, can someone show me exactly how this is done? ...
1
vote
0answers
101 views

Sum of natural numbers

$$1+x+x^2+x^3+x^4+...=\sum_{k=0}^{\infty} \left(x^k\right)=\frac{1}{1-x}, |x|<1$$ $$\frac{d}{dx} \left(x^n\right)=nx^{n-1}\Longrightarrow$$ $$1+2x+3x^2+4x^3+5x^4+...=\sum_{k=0}^{\infty} ...
0
votes
0answers
80 views

sum and infinity

If you have the sums $ (1+2+..+n) + (1+2+3+..+n-1)+ (1+2+3+..+n-2)+(1+2+3+..+n-3)+...+(1+2+3)+(1+2)+1$for large enough $n$ $$\frac {n^3}{3!} \approx (1+2+..+n) + (1+2+3+..+n-1)+ ...
2
votes
2answers
648 views

True or false identity?

I found the logo from The Eighth Congress of Romanian Mathematicians. I think this is the von Mangoldt summatory function and with a simple computation, using this definition, I obtained $83$. Am I ...
1
vote
2answers
23 views

Is there a way to turn this summation into a matrix multiplication?

I have two vectors $\mathbf{s}, \mathbf{p}$ of length $n$, and I need to compute a vector $\mathbf{\pi}$ defined by $$\pi_i=\sum_{j=1}^is_j(p_j-p_i)$$ for $i$ from $1$ to $n$. I suspect this ...
4
votes
2answers
68 views

$\sum_{n=1}^{\infty} \frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)$ diverges [on hold]

How can I prove the divergence of the series $$\sum_{n=1}^{\infty} \left(\frac{1}{(n+1)!} \prod_{k=1}^{n} f(k)\right) $$ if $f:\mathbb{N} \rightarrow \mathbb{N}$ is injective? $ $
0
votes
2answers
66 views

Summation of special series

Does anybody know how to evaluate $$\sum_{i=2}^n(i^2)\cdot{i\choose2}$$ How about the general case of $(i^k)*{i\choose2}$? A nice formula would be great!
-1
votes
2answers
60 views

Evaluate the sum below [on hold]

Evaluate the following sum $$1*1!+2*2!+3*3!+....+1000*1000!$$ any help guys?
0
votes
0answers
31 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
38 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
23 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
vote
2answers
35 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
387 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
70 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
63 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
144 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
vote
2answers
103 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
votes
0answers
19 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
votes
6answers
122 views

What is the mathematical symbol for the sum of numbers

For example, when $n=5$, what is the symbol for $5+4+3+2+1$?
2
votes
0answers
54 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
221 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
123 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 ...
1
vote
3answers
102 views

Binomial Sum: Values

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
votes
4answers
94 views
+100

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
58 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
64 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
votes
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
votes
0answers
22 views

The Summation of a Summation in Mathematica [closed]

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
25 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 ...
9
votes
2answers
213 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
74 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
473 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
votes
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
vote
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 ...