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

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5
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4answers
67 views

Interesting summation question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to?

Question: If $a$ and $b$ are the roots of $x^2+x+1$, then what is the below expression equal to? $$\sum_{n=1}^{1729} \left[(-1)^n\cdot V(n)\right]$$ Where $$V(n)=a^n+b^n$$ My effort: I think I ...
2
votes
3answers
46 views

Continuity of function consisting of an infinite series.

Let $f(x) , 0\leq x\leq 1$ be defined by, $$f(x)=\sum_{n=1}^{\infty}\frac{1}{(x+n)^2}$$. Show that $f$ is continuous on $[0,1]$ and that, $$\int_0^1f(x)dx=1$$. I have never dealt ...
2
votes
3answers
65 views

Calculating $\sum_{k=0}^{n}\sin(k\theta)$ [duplicate]

I'm given the task of calculating the sum $\sum_{i=0}^{n}\sin(i\theta)$. So far, I've tried converting each $\sin(i\theta)$ in the sum into its taylor series form to get: ...
1
vote
2answers
46 views

Is $\sum_{n=1}^\infty a_n\sin(nx)$ converges on $[\varepsilon, 2\pi-\varepsilon]$?

Let $a_n$, a sequence monotonically decreasing to $0$. Consider $$\sum_{n=1}^\infty a_n\sin(nx)$$ Is the series converges uniformly on $[\varepsilon, 2\pi-\varepsilon]$? ($\varepsilon ...
0
votes
1answer
39 views

Multiplication of 2 sums that equal another multiplication of 2 sums

I have been trying to prove a formula of mine and i come across something very interesting, well to me it is. If the formula is correct, it states that: $$ \left(\sum_{m=0}^{k-c} {k-c \choose m}{ms_1 ...
1
vote
0answers
36 views

How to Evaluate this Summation to Find a Closed Form

While taking the incomplete Bell Polynomil of $x^a$ i found out that: $$ B_{n,k}^{x^a}(x) = x^{ak-n} \sum_{m=0}^k \frac{(am)!(-1)^{k-m}}{m!(k-m)!(am-n)!} $$ Now, what i am wondering is, what is the ...
3
votes
1answer
148 views

Generalized Sophomore's dream. Question about originality

A few months ago I derived a beautiful fact: $$ \sum_{n=k+1}^\infty n^{k-n}=\int_{0}^{1} t^{k-t}dt~~~(*) $$ for every natural $k$. Generally: $$ \sum_{n=1}^\infty ...
2
votes
1answer
46 views

How do I calculate these sum-of-sum expressions in terms of the generalized harmonic number?

I know that $$\sum_{m=2}^k\sum_{n=1}^{m-1}(nm)^{-s}=\frac 12((H_k^s)^2-H_k^{(2s)})$$ and $H_k^s=\sum_{n=1}^kn^{-s}$ But, how would I go about finding identities in terms of the harmonic number like ...
0
votes
1answer
31 views

Formula for $\sum_{i = 1}^n k^n$ [duplicate]

I know from my calculator the answer is $\sum_{i = 1}^n k^n$ = $\frac{k^{n+1}-k}{k - 1}$. I'd just like help understanding why.
4
votes
2answers
81 views

Showing $\sum_{n=1}^\infty \sin x \sin nx$ is uniformly bounded

I need to show that for every $x$: $$\sum_{n=1}^\infty \sin x \sin nx \lt M$$ So the first thing came into my mind is applying a well-known trigonometric identity: $$\sum_{n=1}^\infty \sin x \sin nx ...
2
votes
3answers
73 views

value of an $\sum_3^\infty\frac{3n-4}{(n-2)(n-1)n}$

I ran into this sum $$\sum_{n=3}^{\infty} \frac{3n-4}{n(n-1)(n-2)}$$ I tried to derive it from a standard sequence using integration and derivatives, but couldn't find a proper function to describe ...
4
votes
1answer
77 views

Find the remainder when the sum is divided by $1000$

Find $S \pmod{1000}$ given: $$S = \sum_{n=0}^{2015} n! + n^3 - n^2 + n - 1$$ $$S_0 = 0! + 0 - 0 + 0 -1 = 0$$ $$S_1 = 1! + 1 - 1 + 1 - 1 = 1$$ $$S_2 = 2! + 8 - 4 + 2 - 1 = 7$$ This isn't ...
1
vote
1answer
42 views

How to Split a Sequence of Numbers Into Four (Relatively) Equal Summations

How would I go about splitting a sequence of numbers into four equal (as equal as possible) summations? Say I have a sequence of 26 integers like so: 16, 4, 17, 10, 15, 4, 4, 6, 7, 14, 9, 17, ...
0
votes
1answer
32 views

How to evaluate this combination of sums and integrals?

I am reading a book on PDEs, and I am near the beginning where the author is talking about the heat equation and, specifically, solving the non-homogenous equation $u_t={\alpha}^2u_{xx}+f(x,t).$ The ...
1
vote
1answer
76 views

Sum of trigonometric infinite series

I am trying to prove that for any $x\geq 1$ we have: $$ \sum_{m=1}^{\infty} \frac{\sin\frac{(2m-1)\pi}{x}}{\left(\frac{(2m-1)\pi}{x}\right)^3} = \frac{x}{8}(x-1). $$ Could I have some help please? I ...
2
votes
2answers
22 views

Finite double sum: Improve index transformation

In order to prove a rather complicated binomial identity a small part of it implies a transformation of a double sum. The double sum and its transformation have the following shape: ...
0
votes
3answers
97 views

Sum of $\sum\limits_{x=-\infty}^{\infty}x^{\operatorname{sign}(x)}$

Both the sum of $1+2+3+4+\cdots$ and the sum of $\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+\frac{1}{5}+\cdots$ diverge. If both are paired together in one function, as seen above, can they amount to a ...
1
vote
2answers
66 views

How to prove that $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$

I am currently trying to prove: $\lim\limits_{n \to \infty} \sum\limits_{k = 1}^{n} {\sqrt k \over n^{1.5}} = {2 \over 3}$ I can easily squeeze the series between 0 and 1. I don't know many handy ...
6
votes
2answers
62 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 $$\sum_{k=1}^n \sum_{a_1+a_2 + \cdots +a_k=n} \binom{n}{a_1,a_2, \cdots , a_k} \binom mk \binom{k}{b_1,b_2, \cdots , b_l}= m^n,$$ ...
2
votes
3answers
77 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
27 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
47 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
98 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
101 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
32 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
46 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
56 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
29 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
1answer
48 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
103 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
91 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
655 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
25 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 [closed]

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
67 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
62 views

Evaluate the sum below [closed]

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
388 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
71 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
64 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
146 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
106 views

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

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
124 views

What is the mathematical symbol for the sum of numbers

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