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
34 views

Fourier Series Representation of $t + 1$

I need your help double-checking my working for finding the Fourier Series representation of a "piecewise" function. The function I was given is $$f(t) = \begin{cases}t + 1 & ,-1 < t \le 1 \\ ...
0
votes
2answers
54 views

What is the sum of binomial-coefficients multiplied by i?

$$\sum_{i=0}^n {n \choose i}i$$ I know this is equivalent to $\sum_{i=0}^n \frac{n!i}{i!(n-i)!}$, but the factorial prevents me from solving this easily.
2
votes
6answers
208 views

Summing the terms of a series [closed]

I have a really confusing question from an investigation. It states- Find the value of: $$\sqrt{1^3+2^3+3^3+\ldots+100^3}$$ How would I go about answering this??
3
votes
0answers
49 views

The $n$th integral of $\ln(x)$ and fractional derivatives

For a related question, I need to know the $n$th integral of $\ln(x)$ and the fractional derivative of $\ln(x)$. A break down of how fractional derivatives may be found on the Wikipedia. In ...
2
votes
0answers
50 views

I am looking for a disproof of this conjecture on closed forms

Assume $I=\int_0^\infty f(x)\text{ d}x$ and $J=\sum_{n=0}^\infty f(n) ; I,J\in\Bbb{R}$ Conjecture: If $I$ has a closed form, then $J$ must carry a closed form. Can someone find a proof or ...
1
vote
1answer
82 views

Closed form solutions to a recursion relation

Consider a following recursion relation of degree two: \begin{equation} y_{n+1} = f_n \cdot y_n + g_n y_{n-1} \end{equation} for $n\ge 1$ subject to $y_1={\mathcal F}_1$ and $y_0={\mathcal F}_0$. By ...
0
votes
1answer
53 views

Find a generating function with Fibonacci

$$G(x) = \sum_{n=1}^\infty na_n x^n $$ Hello. I need to find a generating function for the summation above, where $a_n$ is the Fibonacci sequence. I have found the generating function for the ...
5
votes
5answers
339 views

summation of binomial coefficients with squares

What is $$50^2\frac{{n\choose 50}}{{n\choose 49}}+49^2\frac{{n\choose 49}}{{n\choose 48}}...1^2\frac{{n\choose 1}}{{n\choose 0}}$$. i.e. $$\sum_{k=1}^{50} \frac{k^2\binom n k}{\binom n {k-1}}= ?$$ ...
33
votes
3answers
2k views

Does the sum of the inverses of the sums of the primes converge?

$$\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n} = \frac{1}{2} + \frac{1}{5} + \frac{1}{10} + \frac{1}{17} ... $$ Where $p_n$ is the $n$th prime number, does $\sum_{m=0}^∞ \frac{1}{\sum_{n=0}^m p_n}$ ...
7
votes
1answer
54 views

Is the sum of the square roots of all natural numbers up to n whole for any value of n other than 1?

For the summation $\sum_{n=0}^x \sqrt{n}$ are there any values of $x$ where the summation equals a whole number other than 1?
0
votes
1answer
31 views

a specific function applied over a finite sequence of natural numbers

First, i'm not a mathematician, and also English is not my first language, so if this question is a duplicate or an equivalent of an existing one - its due to my unsuccessful searching attempts. I ...
2
votes
2answers
48 views

What is the asymptotic behaviour of $\sum_{p_k\leq x}kp_k$, where $p_k$ is the kth prime number?

I would like to study the asymptotic behaviour of this sequence A014285, see as OEIS, that seems has few references and a good behaviour (see the sequence as graph) $$\sum_{k=1}^nkp_k,$$ where $p_k$ ...
1
vote
1answer
64 views

Find the value of n if:

$$\sum_{k=0}^n (k^{2}+k+1) k! = (2007).2007!$$ How to approach this problem? In need of ideas. Thank you.
2
votes
4answers
87 views

why $(\sum 8(n-1)) +1$ is equals to $(2n-1)^2$

(Forgive me if it is a silly question) When I was solving a puzzle, I observed a sequence 1, 1+8, 1+8+16, 1+8+16+24, 1+8+16+24+32.... is equals to ...
4
votes
0answers
58 views

Finite sum with three binomial coefficients

I need to find a closed form expression, if there is one, of the following sum: $$\sum_{j=0}^m{n+1-k\choose j}{k-1\choose m-j}{A+2-k+m-j\choose m-j+2}$$ where all parameters are integers, $~1\leq ...
-2
votes
6answers
216 views

Simplify $\binom nr+ 4\binom n{r+1} + 6\binom n{r+2} + 4\binom n{r+3}+\binom n{r+4}$ [closed]

For $$4\le r \le n,$$ $${n \choose r}+4{n \choose {r+1}}+6{n \choose {r+2}}+4{n \choose {r+3}}+{n \choose {r+4}}$$ equals 1. $${n+4}\choose{r+4}$$ 2.$${n+4}\choose{r}$$ 3.$${n+3}\choose{r-1}$$ ...
0
votes
2answers
21 views

Summation Simplification Confusion

$$(1/2)\sum_{i = 0}^{n/2 - 1} i - \sum_{i = 0}^{n/2 - 1} 1$$ Would this require the use of the following identitiy for the first summation: $$ \frac{n(n+1)}{2} $$ I attempted to simplify ...
1
vote
4answers
28 views

Can I change this summation to a sum of other summations?

The form of the summation I have is $$\sum _{ x=0 }^{ \infty }{ x{ a }^{ x } } $$ I need to somehow remove the $x$ from the original summation in order to achieve the geometric series in each other ...
0
votes
2answers
28 views

Simplifying Summation

$$\sum_{i = 0}^{n/2 - 1} n = n^2/2 - n$$ A quick question on summation simplification that is not clear to me. In this case is multiplying $n$ by $n/2 -1$ valid? Or does it need to be plugged ...
1
vote
0answers
14 views

Change of basis to a reduced set when function has known values

Let $\gamma_1\left(x\right) = \sum_{n=1}^{N} \alpha_n f_n\left(x\right)$ where $f_n\left(x\right)$ are orthonormal under an inner product over a range $\left[x_1, x_2\right]$. I already know all of ...
2
votes
2answers
23 views

Find the Laurent series for $p(4/z)$

Find the Laurent series for $p(4/z)$ given that $p(z)=(z-3)^3$ My attempt: if the Taylor series for $p(z)$ looks like $$\frac{-27}{0!}+ \frac{27z}{1!}-\frac{18z^2}{2!}+ ...
2
votes
1answer
26 views

Find the laurent series for $e^{2/(z-1)}$

I'm starting to learn about Laurent series. The way I understand it is that it is the same as a Taylor expansion, but with negative terms in addition to the positive terms. I may be wrong, but isn't ...
4
votes
2answers
54 views

Find the value of $\sum_{n=2}^{\infty}\log\left(1-\frac{1}{n^2}\right)$

Find the value of $$\sum_{n=2}^{\infty}\log\left(1-\frac{1}{n^2}\right)$$ I tried expressing the sum like $\sum a_r-a_{r-1}$. ...
5
votes
1answer
56 views

Confused about infinite sum $\sum\limits_{a,b,c}\frac{a+b+c+abc} {2^a(2^{a+b}+2^{b+c}+2^{a+c})}$

$$ \displaystyle \sum^{\infty}_{a=0} \displaystyle \sum^{\infty}_{b=0} \displaystyle \sum^{\infty}_{c=0}\dfrac{a+b+c+abc} {2^a(2^{a+b}+2^{b+c}+2^{a+c})}= \ ? \ $$ I calculated its value as ...
3
votes
2answers
63 views

General strategies for evaluating sums in probabilistic/combinatoric problems: $\sum_n\sum_m {n+m \choose n}p^{n+m}(1-p)^{n+m}$

I have encountered the following summation: $$p^2\cdot \sum_{n=0}^\infty\sum_{m=0}^\infty {n+m \choose n}p^{n+m}(1-p)^{n+m}$$ This summation arises from naive analysis of this simple probability ...
0
votes
0answers
36 views

On partial summation and the function $\frac{1}{(\Gamma(x+1))^s}$

I've tried follow the know exercise in which we can get $\zeta(s)$ in an integral representation, for $\Re s>1$, combining partial summation and taking the limit $$\lim_{x\to\infty}\sum_{n\leq ...
-1
votes
3answers
61 views

How to evaluate the finite sequence (involving the floor function)? [closed]

How to sum this:$$ \lfloor\sqrt{1}\rfloor + \lfloor\sqrt{2}\rfloor + \lfloor\sqrt{3}\rfloor +\lfloor\sqrt{4}\rfloor + \ldots + \lfloor\sqrt{50}\rfloor = ?$$
2
votes
2answers
85 views

How is Faulhaber's formula derived?

I have been wanting to understand how to find the sum of this series. $$1^p + 2^p + 3^p +{\dots} + n^p$$ I am familiar with Gauss' diagonalised adding trick for the sum of the first $n$ natural ...
0
votes
1answer
33 views

I want know how to get $\sum_{m \gt 0, m\neq n } \frac{1}{m-n}- \frac{1}{m+n}=\frac{3}{2n}$

I am trying to understand this summation: $$\sum_{m \gt 0, m\neq n} \frac{1}{m^2-n^2}=\frac{1}{2n} \sum_{m \gt 0, m\neq n } \left(\frac{1}{m-n}- \frac{1}{m+n}\right)=\frac{3}{4n^2}$$ But I can't see ...
1
vote
1answer
45 views

Finite derivative of the harmonic series

In Knuth's Concrete Mathematics he represents the famous quicksort algorithm in computer science as a infinite sum then shows that sum can be simplified to being essentially harmonic. I want to ...
0
votes
1answer
31 views

Understanding “divides” notation, aka “|”, in sigma notation

Sigma notation in question Hello, In the picture above, there is a $d \mid (k,n)$ under the sigma notation. I know this means that $d$ has to divide the highest common factor of $k$ and $n$, but I ...
0
votes
0answers
49 views

consider the function $f(1) = 1$, $f(n) = \sum_{i = 1}^{n - 1}(if(i))$ for $n > 1$

Consider the function $f(1) = 1$ $f(n) = \sum_{i = 1}^{n - 1}(if(i))$ for $n > 1$ Let $A(n)$ be the worst-case number of scalar arithmetic operations (+,-,*,/) required by this function for ...
0
votes
0answers
14 views

Understanding $ \mu$ notation as it applies to $ \mu(d)$

Paragraph in question. Hello, I'm wondering about the $\mu(d)$ notation used with the sigma notation in the paragraph above. I don't know what it refers to. I understand all the other symbols used in ...
3
votes
1answer
53 views

A problem with the Legendre/Jacobi symbols: $\sum_{n=1}^{p}\left(\frac{an+b}{p}\right)=0$ [duplicate]

This problem is taken from Niven's textbook, 3.6.16. Prove that if $(a,p)=1$ and $p$ is an odd prime, then $\sum_{n=1}^{p}\left(\frac{an+b}{p}\right)=0$, where $\left(\frac{x}{y}\right)$ is the ...
4
votes
4answers
117 views

How to calculate $\sum_{m=0}^{i} (-1)^m\binom{2i}{i+m}=\frac{1}{2}\binom{2i}{i}$

how can we calculate this?$$ \sum_{m=0}^{i} (-1)^m\binom{2i}{i+m}=\frac{1}{2}\binom{2i}{i} $$ It is alternating and contains the Binomial coefficients which are given in terms of factorials as, $$ ...
4
votes
3answers
85 views

How to derive this binomial identity?

I believe the following is an identity (I've tested with a few random $m$ and $n$ values, could be wrong though): $$\sum_{k= 0}^{\infty}{m \choose k}{n \choose k}k=n\binom{m+n-1}{m-1}$$ but I'm not ...
4
votes
2answers
141 views

What is this binomial sum?

I'm trying to figure out what this sum is equal to: $$\sum^n_{k=0}k \binom{m-k}{m-n}$$ I thought there are n turns and on each turn you pick 1 object from k objects ($\binom{k}{1}=k$) and also pick ...
3
votes
6answers
135 views

Proof that the sum of the first $n$ odd numbers is $n^2$.

Here is what I have so far: The $n$th odd number is $2n-1$. So we prove that $1+3+...+(2n-3)+(2n-1)= n^2$. Separate the last term and you get: $[1+3+...+(2n-3)]+(2n-1)$ $[1+3+...+(2n-3)]$ is the ...
5
votes
2answers
118 views

Find a sum $S = \sum\limits_{t \in E} \sum\limits_{x \in E} (t + x)(t + x^2)…(t+x^{2p})$

I am solving a problem that has already a plan for the solution. As a subproblem I have to find the value of $$S = \sum\limits_{t \in E} \sum\limits_{x \in E} (t + x)(t + x^2)...(t+x^{2p})$$ where ...
5
votes
4answers
118 views

How to prove the following binomial identity

How to prove that $$\sum_{i=0}^n \binom{2i}{i} \left(\frac{1}{2}\right)^{2i} = (2n+1) \binom{2n}{n} \left(\frac{1}{2}\right)^{2n} $$
3
votes
2answers
82 views

What is the expanded form of $\sum_\limits{0}^{0}{f(x)}$?

The is a MCQ in my math book which says the following: Expanded form of $\sum\limits_0^0{f(x)}$ is: 1) $0$ 2) $f(0)$ 3) $1$ 4) None I don't know which one is correct but one of the first ...
0
votes
0answers
18 views

How can we prove that $\sum_{i=1}^n\cos\left(i\pi\frac kn\right)=0$, if $k\in2\mathbb N\setminus\left\{0\right\}$ [duplicate]

How can we prove that $$\sum_{i=1}^n\cos\left(i\pi\frac kn\right)=\begin{cases}n&\text{, if }k=0\\ 0&\text{, if }k\in2\mathbb N\setminus\left\{0\right\}\\-1&\text{, if }k\in2\mathbb ...
1
vote
2answers
24 views

summation related to reciprocals

I know that $\sum\frac{1}{n^2}=\frac{\pi^2}{6}$ in a test this was given and we were asked to find $\sum \frac{1}{(2n+ 1)^2}$ starting with $n=0...\infty$. Now i am grade $11$ student and have been ...
4
votes
3answers
131 views

Is there a closed-form expression for this nested sum?

Is there a closed-form expression for this nested sum? $$s(n)=\sum_{i=1}^n\;\; \sum_{j=i+1}^n \sum_{k=i+j-1}^n1$$ If yes, what is it and how can it be derived?
1
vote
2answers
50 views

Convergence of Infinite Sum [closed]

Using Mathematica, I have been able to make the following statement based on numerical evidence: $$\sum_{i=0}^\infty \frac{2^i}{x^i}=\frac{x}{x-2}$$ for any $x≥3$. How can this be proven?
0
votes
1answer
23 views

Proof two index sets are equal

I'm having trouble proving these two sums are equal with any change of variables, I must be missing something obvious but cannot see it. Any comments appreciated. $$ \sum_{m=1}^{L-1} ...
1
vote
1answer
16 views

Summation multiplication

What is wrong with writing: $\displaystyle \Sigma_{n}\ a_n .$ $\Sigma_{n}\ b_n$ ? I understand that it does not matter what dummy variable you sum over, but I don't understand why this is seen as ...
1
vote
3answers
64 views

Prove that a specific inequality holds

Let $n \in \mathbb{N}$. Let $z_1, \ldots, z_n$ and $w_1, \ldots, w_n$ be complex numbers such that $$ \sum_{j = 1}^n |w_j|^2 \leq 1 $$ and $$ \left| \sum_{j = 1}^n z_j w_j \right| \leq 1 $$ Show that ...
1
vote
1answer
28 views

Infinite sum of discrete unit-step signals

Trying to sketch the following signal: $$\sum_{k=-\infty}^\infty (u[k]-u[k-3])(u[n-k]-u[n-k-3])$$ Where $u[n]$ is the unit step signal (the Heaviside function, $1$ when $n\ge 0$ and $0$ otherwise). ...
4
votes
1answer
24 views

When Is the Remainder Less Than The Quotient

I was given the problem: Let $f(n)$ be the number of times $a$ is $n$-well for $1\le a \le n$. An integer, $a$, is $n$-well if $$\left\lfloor\frac{n}{\left\lfloor ...