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

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3answers
98 views

Sum of powers: $1^m+2^m+3^m+…+n^m$=? [closed]

For any positive integer $n$ and $m,$ I was wondering if there is any way to get a closed formula for $$S(n,m)=1^m+2^m+3^m+\cdots+n^m$$ something like $$S(n,1)=1+2+3+\cdots+n=\frac{n(n+1)}{2}.$$
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2answers
48 views

How to represent the following in sum notation,also notify if this is any special series?

$\color{blue}{1.}~~ -x+\frac{x^3}{3}-\frac{x^5}{5}+\frac{x^7}{7}\cdots \infty$ and $\color{blue}{2.}~~ \frac{x^2}{2}-\frac{x^4}{4}+\frac{x^6}{6}-\frac{x^8}{8}\cdots \infty$ i know $1,3,5,7\cdots$ ...
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1answer
50 views

prove that $\lim \limits_{n \rightarrow \infty} n \sum \limits_{j=1}^{n} \frac{cos(\frac{n}{j}) f(\frac{n}{j})}{j^2}$ exists and final.

$f$ is monotonically decreasing function such that $\lim \limits_{x \rightarrow\infty} f(x) =0$, prove that the following limit exists and final . $$\lim \limits_{n \rightarrow \infty} n \sum ...
4
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0answers
75 views

Calculation of an expression ($\max_{U}\min_i \sum_j |U_{ij}|^2 |e_i^j|^2$)

There is an orthonormal basis $\{e_i\}(i=1,\ldots,n)$ in $\mathbb{C}^n$, each of them is represented in form of column vectors $$\begin{pmatrix} e_i^1\\ \vdots\\e_i^n\end{pmatrix}.$$ My purpose is to ...
0
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2answers
54 views

Prove, that the sum difference of all consecutive prime numbers from $p_1$ to $p_n$ is $p_n-p_1$

Example: $\mid (2-3)+(3-5)+(5-7)+(7-11)\mid =11-2=9$ I tried a couple of basic tricks to reach some proof but I failed.
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1answer
43 views

Gradient of a summation

How to calculate the gradient of the following summation in terms of $x_i$ ? $$\sum_{i=0}^n(x_i-a)^2$$ is the following answer true? $$2 \sum_{i=0}^n(x_i-a)$$ Thank you
4
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1answer
90 views

Interchange summation and differentiation for ONB

Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$ Now assume we have that $$\frac{d}{dx}e_n = \lambda_n e_n.$$ Assume $f \in H^1[0,1],$ so i.e. $||f'||_{L^2} < \infty$ ...
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0answers
49 views

Summation by parts with only one variable

I have a question about summation by parts: I am supposed to show that $\sum_{j=1}^{n}(v_j^m)(v_{j-1}^m-2v_j^m+v_{j+1}^m)=-\sum_{j=1}^{n}(v_{j+1}^m-v_j^m)^2$ where m is dependent on time. I am given ...
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1answer
15 views

Confused about a step with the summation operator.

Image 1 contains the step which I am confused about. What happens to the middle term (-XiY(bar) -YiX(bar)). Picture 2 contains the question for context. I understand how to do the question once I ...
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2answers
72 views

Prove, If the sum of the first $n$ prime is also a prime then it is also a hypotenuse of a primitive Pythagorean triples

I checked this for all the primitive Pythagorean triples $<300$. Some examples would be: a. $2+3=5$, b. ...
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2answers
36 views

Find the Harmonic Mean

The question: Peter drives to work, a distance of 50 miles, at a speed of 75 mph and returns home at a speed of 80 mph. What is his average speed for the round trip? The formula to use here is ...
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0answers
18 views

If $s_{k,m}(n) =\sum_{i=n+1}^{kn+m} \frac1{i} $ show that for $k \ge 2m+1$, $s_{k,m}(n+1)>s_{k,m}(n)$ and $s_{k,m}(n+1)-s_{k,m}(n) <\frac1{n(n+1)} $

Let $s_{k,m}(n) =\sum\limits_{i=n+1}^{kn+m} \frac1{i} $. Show that, for $k \ge 2m+1$, $s_{k,m}(n+1)>s_{k,m}(n)$ and $s_{k,m}(n+1)-s_{k,m}(n) <\frac1{n(n+1)} $ so that $s_{k,m}(n) < ...
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1answer
35 views

Application: Sum of Digits

if a five digit number N is such that sum of its digit is 29, can N be square of an integer? Suppose N be abcde, where a+b+c+d+e = 29. Can square of any number less than abcde is equal to abcde ...
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2answers
47 views

How to deduce sum's result?

I was solving some tricky-task with algorithms and I obtained following reccurence: $$a_{k+1} = 4a_{k} + 16^{k}, a_{1} = 1$$ It's obvious that with given start condition: $$a_{k+1} = 4a_{k} + 16^{k} ...
0
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4answers
38 views

How to computer the summation of a binomial coefficient/ show the following is true

$\sum\limits_{k=0}^n \left(2k+1\right) \dbinom{n}{k} = 2^n\left(n+1\right)$. I know that you have to use the binomial coefficient, but I'm not sure how to manipulate the original summation to make ...
2
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3answers
171 views

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$

Find the sum $\sum_{n=1}^{50}\frac{1}{n^4+n^2+1}$ $$\begin{align}\frac{1}{n^4+n^2+1}& =\frac{1}{n^4+2n^2+1-n^2}\\ &=\frac{1}{(n^2+1)^2-n^2}\\ &=\frac{1}{(n^2+n+1)(n^2-n+1)}\\ ...
1
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2answers
52 views

Probability that colored balls are separated

Say we throw $b$ blue balls and $r$ red balls uniformly into $n$ boxes. The probability that no box contains a red as well as a blue ball is then, by the inclusion exclusion principle: $$p = ...
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1answer
131 views

Inclusion relation between two summability methods

Let $0\leq x<1$ and $s_n$ be a sequence of partial sums of the series $\sum_{n=0}^{\infty}a_n$. It is called that the series $\sum_{n=0}^{\infty}a_n$ is $(A)$ or Abel summable to $s$ if ...
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2answers
84 views

Closed Form for Finite Sum: Product of two Similar Functions

I need to find a closed form expression in terms of $c$, $n$, $x$ and $y$ for $$ \sum_{j=0}^{n}\rho^{c-j}\frac{x^j}{j!}\frac{y^{c-2j}}{\left(c-2j\right)!} $$ where $c$ and $\rho$ are just constants. ...
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2answers
53 views

Reciprocal squares sum inequality [duplicate]

What is the easiest (preferably inductional) way without approximation of the sum_ to prove the following inequality: $\frac{1}{1^2}+\frac{1}{2^2} + \ldots +\frac{1}{n^2} \le 2 - \frac{1}{n}$
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0answers
39 views

Sum of exponents equals delta function

For a homework exercise i need to prove that $$\sum_{n\in\mathbb{Z}} e^{i(k-\xi)n}|u(\xi)> =\delta(k-\xi)|u(\xi)>.$$ Where $\delta(x)=1$ if $x=0$ and zero elsewhere and $|u(\xi)>$ is just a ...
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0answers
22 views

Question regarding Einstein summation convention:

I have this homework where I have to prove long vector identities using Einstein summation convention. This is straightforward if you just follow the rules, however, if you want to keep track of why ...
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0answers
21 views

Combinatorics: Verify the identity: $\sum_{k=0}^{n}$ $\frac{(2n)!}{(k!)^2((n-k)!)^2} = {2n \choose n}^2$ [duplicate]

Verify the identity: $\sum_{k=0}^{n}$ $\frac{(2n)!}{(k!)^2((n-k)!)^2} = {2n \choose n}^2$
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3answers
39 views

$N =\sum_{k = 1}^{1000}k(\lceil\log_{\sqrt{2}}k\rceil-\lfloor\log_{\sqrt{2}}k\rfloor). $

Find $N$ for $$N =\sum_{k = 1}^{1000}k\left(\left\lceil\log_{\sqrt{2}}k\right\rceil-\left\lfloor\log_{\sqrt{2}}k\right\rfloor\right)\;.$$ How could you solve this problem? Are there sigma rules or ...
1
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1answer
86 views

On the $n^{th}$ day he puts $n$ pennies into the same jar. Which day is the first day on which he has at least $20$ dollars in the jar?

On the first day, Daniel puts one penny into the jar. On the second day he puts $2$ pennies into the same jar. On the $n^{th}$ day he puts $n$ pennies into the same jar. Which day is the first day ...
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1answer
23 views

Summation to the square of n -n

Does anyone know the correct way to write a summation starting with (n^2)-n subtracting n, ending at n ... I'm trying to find out if it's legitimate to put (n^2)-n as the lower limit, or to put n ...
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4answers
39 views

Convergence of $\sum_{n=1}^\infty\frac{3}{2n^{p+1}}$

Convergence of $\sum_{n=1}^\infty\frac{3}{2n^{p+1}}$ . What will be the best proof for convergence of this series, which criterion will be the best?
1
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1answer
37 views

Why do O(logn) & O(exp(n)) Have Polynomial & Non-Polynomial Running Time Complexities Respectively Despite Their Taylor Series?

I understand that a function, say $f(x)$, belongs to a class $O(g(x))$ iff: $$ \exists k > 0 \ \ \exists \ \forall n > n_0: |f(n)| \leq |g(n) \cdot k| $$ I also know that $log(x)$ is has ...
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0answers
17 views

Mutual difference of vectors squared, does it have a name?

Given a set of $n$ vectors $\def\vv{\vec{v}} \vv_i$ with the additional property that they all have the same absolute value $||\vv_i||=c$, define the average of the vectors as $\vv = ...
1
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1answer
30 views

Proof of summation formula

I'm trying to help my son with his A level maths. He has the following problem: i) Prove: $$\sum_{r=1}^n \{(r+1)^3 - r^3\} = (n + 1)^3 - 1$$ ii) Prove: $$(r + 1)^3 - r^3 = 3r^2 + 3r + 1$$ iii) ...
2
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2answers
40 views

Prove the sum of the even Fibonacci numbers

Let $f_n$ denote the $nth$ Fibonacci number. Prove that $f_2\:+\:f_4\:+...+f_{2n}=f_{2n+1}-1$ I am having trouble proving this. I thought to use induction as well as Binet's formula where, ...
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1answer
45 views

Using Partial Summation to evaluate a series

$$S = \sum_{x=1}^{\infty} \frac{\sin(x)}{x}$$ Using partial summation. Obviously, $$S = \lim_{n \to \infty} \sum_{x=1}^{n} \frac{\sin(x)}{x}$$ Partial Summation: \begin{align*} \sum_{n=1}^{N} ...
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2answers
48 views

Is joining split sums as a nested sum just the distributive property? How to prove it?

I'm studying tensors right now and there are lots of sums involved and I've seen they do these sorts of things: $(\sum_i x)(\sum_j y) (\sum_k z)=\sum_k \sum_j \sum_i x y z$ I've shown mechanically ...
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3answers
106 views

Rewriting exponents as sigmas, is this a thing?

Is this a thing? If not, can anyone help me out on this? So I saw this a while back.$$\sum_{i=0}^n(2n + 1) = n^2$$ For positve n. This is interesting, and I wondered if I could write any n^x in ...
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0answers
20 views

Distribution of modulus of sum of complex numbers with known moduli

If a finite set of complex numbers have known moduli but uniformly distributed arguments, what is the distribution of the modulus of their sum? (The problem arises from the practical problem of ...
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3answers
62 views

Do we sum over sets?

I'm writing my thesis on a topic in machine learning. In some parts I defined a summation over a set. I sent my thesis draft to my professor and then I met him. He briefly mentioned that "we don't sum ...
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4answers
48 views

Prove Binom Sum $\sum_{k=0}^n(-1)^k \binom{n}{k} = 0$ [duplicate]

Let: $$ (-1)^0=1 $$ I need to prove that: $$ \sum_{k=0}^n(-1)^k \binom{n}{k} = 0 $$ Thanks!
3
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2answers
84 views

How to solve the limit $\lim_{n\to\infty}\sum_{k=1}^{n} \frac{k}{k\,n+2n^2}$

I think it is related to squeeze theorem, but could not come up with a solution. The answer here is $1-\ln(9/4)$. Can someone help me with this question? $$ \lim_{n\to\infty}\sum_{k=1}^{n} ...
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0answers
30 views

on the sum of ordered log functions

I have the following question and I need some suggestions on how to address it. Assume we have the following non-increasing ordered positive constants (not variables) $a_i, i = 1, ..., N,$ (i.e., we ...
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1answer
49 views

Hint to solve the following summation identity

Let $\overline{u}$, $\overline{v}$ and $m_{uv}$ be defined as follow: $$\overline{u}=\frac{1}{n} \sum_{i=1}^{n}u_i$$ $$\overline{v}=\frac{1}{n} \sum_{i=1}^{n}v_i$$ $$m_{uv}=\sum_{i=1}^{n}(u_i- ...
0
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1answer
22 views

Algebraic manipulation of product of two summations

If $F_1(x)=\sum_{n\ge 0}a_nx^n$, $F_2(x)=\sum_{n\ge 0}b_nx^n$, why does the following hold? $$\begin{align*} F_1(x)F_2'(x)&=\sum_{n\ge 0}a_nx^n\sum_{n\ge 0}nb_nx^{n-1}\\ &=\sum_{n\ge ...
2
votes
3answers
78 views

General term of the series - find

What is the general term of the series: $$-\frac{1}{2}-\frac{1}{3}+\frac{1}{4}+\frac{1}{5}-\frac{1}{6}-\frac{1}{7}+...$$ I think that the denominator will be $(n+1)$. But what next?
1
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2answers
124 views

Logarithmic Integral Inequality

Give a convincing argument that the following inequalities are true: $$\int_1^n \log x\mathrm dx \leq \log1 + \log2 + ... \log n \leq \int_1^{n+1}\log x \mathrm dx$$ for any $n \geq 1 $ . We are given ...
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0answers
17 views

Showing $\sum_{a = 1}^{A}\max_{b = 1, 2, \ldots, B}f(a, b) = \max_{b = 1, 2, \ldots, B}\sum_{a = 1}^{A}f(a, b)$

Let $f: \mathbb{N} \rightarrow \mathbb{R}_{> 0}$ and $A, B$ are positive integers $ \geq 1$. Is it true that $$\sum_{a = 1}^{A}\max_{b = 1, 2, \ldots, B}f(a, b) = \max_{b = 1, 2, \ldots, B}\sum_{a ...
0
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0answers
20 views

Changing the index of the sums when changing the sums - why this way?

Let $(X_n)_{n\in\mathbb{N}_0}$ be a Markov chain with state space $E$. For $i,j\in E$ set $$ h_i(j):=\mathbb{P}_i(H(j)<\infty):=\mathbb{P}(H(j)<\infty|X_o=i), $$ where $H(j)\colon ...
2
votes
2answers
71 views

How do I find a closed form for this sequence?

I want to find a closed form for $$\sum_{i=0,1,...}{\left\lfloor\frac{n}{2^i}\right\rfloor}$$ e.g. ...
0
votes
5answers
41 views

Find general formula for a series

Having the following series: $$ - \frac{1}{2}+\frac{1}{6}- \frac{1}{10}+\frac{1}{14}-\frac{1}{18}+\ldots$$ What is the easiest approach to find a general formula for this series?
1
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1answer
24 views

Rewriting log of a sum

Suppose we have a vector $$X=[x_1,x_2,\ldots,x_n],\quad x_i\in \mathbb{R} \text{ for } i=1,2,\ldots,n $$ Now if we have a formula $$f_X(x)=\log\left(\sum\limits_{i=1}^nx_i\right)$$ Is it possible to ...
-2
votes
1answer
37 views

Recovering the coefficients $b_r$ of the binomial sum $\sum_{r=0}^n\binom{n}rb_r$ [closed]

Suppose that the sequences of real numbers $a_0,a_1,a_2,a_3,\ldots$ and $b_0,b_1,b_2,b_3,\ldots$ satisfy the relation $$a_n=\sum_{r=0}^n\binom{n}rb_r\;.$$ Then prove that ...
0
votes
3answers
86 views

Evaluate $\sum\limits_{k=1}^{n} \frac{k}{2^k}$ [duplicate]

Evaluate $$\sum\limits_{k=1}^{n} \frac{k}{2^k}$$