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

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3
votes
1answer
65 views

$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$ Identity involving Liouville Lambda function

I have to prove $$\sum_{n=1}^N\lambda(n)[N/n]=[\sqrt{N}]$$ I tried using the approach in this question but I don't know how I'll get $\sqrt{N}$. Please help.
0
votes
1answer
25 views

Formula for weighed geometric sum

I'm trying to find an easy way to derive a formula for: $S_{n} = \frac{1}{n}\sum_{i=0}^{n}(n-i)x^{i}$ I've found a recurrence relationship of sorts: $S_{n+1} = \frac{xnS_{n}+n+1}{n+1} = ...
4
votes
1answer
71 views

Trying to sum a finite series

I've tried to solve a financial mathematical task and I need to sum such a finite series: $$\sum_{k=1}^{6}\frac{1.05^{3k+1}}{1.05^{3k+1}-1}$$ So I decided that maybe I'll try to find a solution to : ...
2
votes
1answer
39 views

Comparing Coefficients in Summations

Suppose I have the following equality: $$\sum_{k=0}^{n-a}\sum_{j=0}^{k}\binom{n}{k}\binom{k}{j}\frac{f(a,k)\cdot g(b,n-k)}{n!}=\sum_{k=0}^{n-a}\binom{n}{k}\binom{n-k}{a}\frac{z^k \cdot ...
4
votes
2answers
271 views

Proof that a sequence that converges to zero has a subsequence whose series is convergent

How would I go about proving that if $a_n$ is a real sequence such that $\lim_{n\to\infty}|a_n|=0$, then there exists a subsequence of $a_n$, which we call $a_{n_k}$, such that $\sum_{k=1}^\infty ...
1
vote
1answer
28 views

Expansion of the square of the sum of $N$ numbers

Do I need to cite any results to use the following equality $$\left( \sum_{n=1}^N a_n \right)^2 = \sum_{n=1}^N a_n^2 + 2 \sum_{j=1}^{N}\sum_{i=1}^{j-1} a_i a_j $$ where $a_n \in ...
1
vote
2answers
65 views

Inequalites between a sum of powers and definite integrals

Give a convincing argument that the following inequalities are true: $$\int_0^n \ x^p\mathrm dx \leq 1 + 2^p + 3^p+ ... + n^p \leq \int_0^{n+1}\ x^p \mathrm dx$$ I've been stuck on this for hours, ...
0
votes
1answer
51 views

Calculate the sum $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$.

I want to calculate this sum: $\sum_{k=0}^{n-1}x^k \binom{n-1}{k} \dfrac{1}{(n-k)!}$. I tried to use some differentation techniques, but they didn't work. Could you help me with this?
1
vote
1answer
75 views

Series of exponential function

I had a thought today and I've tried to see if it is a thing. I'm certain it is a thing, I just don't know how to search for it. We have the Taylor series which is a summation of monomials: ...
0
votes
1answer
40 views

A sum involving a ratio of two binomial factors.

Let $a\ge 0$, $a_1\ge 0$ ,$b \ge 0$ and $b_1\ge0$ be real numbers subject to $1+b+a_1-b_1-a >0$. Let $m$ be a positive integer. Then using methods similar to those in Another sum involving binomial ...
13
votes
13answers
4k views

How to explain the formula for the sum of a geometric series without calculus?

How to explain to a middle-school student the notion of a geometric series without any calculus (i.e. limits)? For example I want to convince my student that $$1 + \frac{1}{4} + \frac{1}{4^2} + ...
0
votes
0answers
26 views

Finite sum over uncountable set

Consider the sum $S=\sum_{x\in I}P(x)$, where $P(x)$ are positive real numbers. When the index set $I$ is finite, $S$ is of course finite. When $I$ is countably infinite, it is also possible that $S$ ...
0
votes
1answer
19 views

A multivariate sum that yields a closed form expression

Let $d\ge 2$ be a an integer. Let $b_1,b_2,\cdots,b_d$ be positive integers. As a by product of certain calculations I have discovered that: \begin{equation} \sum\limits_{q_2=0}^{b_2} \cdots ...
0
votes
1answer
32 views

Building matrix expressions for product of sum, isolating vector of constants

This identity to build the matrix expression for the expression below is pretty straightforward: $$ \left.\sum\limits_{j=1}^M \left( a_j \cdot f_{i,j} \right) \;\right|_{i=1}^N = ...
0
votes
0answers
22 views

closed expression for the sum $ \sum_{n=1}^{\infty} \mu (n)[x/n] $

given the sum $ \sum_{n=1}^{x}\mu (n) [x/n]=f(x) $ is there a closed or asymptotic mode to evaluate this sum ?? thanks here $ [x] $ and 'mu' is th e mobius function
0
votes
0answers
28 views

Integral over all possible paths

May be $f(\vec{x}), \vec{g}(\vec{x})$ an arbitrary functions dependent on the coordinates $\vec{x}=(x,y,z)^T$. Defining the following function dependent on a 3-dimensional curve $\vec{\gamma(t)}$ ...
0
votes
3answers
79 views

Solving a summation for n

I am trying to simplify the following summation: $$\sum_{i=1}^{n/2}\sum_{j=i}^{n-i}j$$ I am not really sure how to solve the inner summation at the moment. I tried this: $$\sum_{j=2i}^{n}j-i$$ and ...
4
votes
2answers
94 views

Let $p_n$ be the $n$th prime, for any integer $n$, prove that: $p_n+p_{n+1}\geq{p_{n+2}}$

I was just wondering about it. True or false, it seems a very interesting question to me. I'm also interested to see how this could be proven or disproven? Opinions are welcome as usual. Regards
1
vote
2answers
53 views

Sum of Products of all Combinations

How to prove that the following expression is true: $$\sum_{i\neq j}^n x_i x_j = \left(\sum_{i=1}^n x_i\right)^2 - \sum_{i=1}^n x_i^2.$$
3
votes
3answers
62 views

Sum with many troubles [duplicate]

I am currently considering a sum $$\sum_{r=0}^{n}{\binom{n}{r} (-1)^{r} (1-\frac{r}{n})^{n}}$$ but have no thoughtful ideas how to start. Maybe it's worth noticing that ...
0
votes
1answer
199 views

Geometric Series Question Given Sum of First 2 and First 3 Terms

The sum of the first two terms of a convergent geometric series is 8 and the sum of the first three terms is 26. What is the sum of the series? I get to $ar^2 = 18$, $r = \sqrt{18/a}$ and then I ...
0
votes
2answers
18 views

Understanding an expression involving sum notation

See the following expression. $$\sum_{i=1}^n \sum_{j \in i} f(a_i)d_j - \sum_{i=1}^n \sum_{j \in i} f(b_j) d_j$$ So in the first two sums, we pick a $i$, and then we sum over all $j$ that satisfy $j ...
3
votes
2answers
68 views

The minimum of the sum

I was trying to prove some theorem on my way. So the problem was: Theorem: Let $x_1, x_2, ..., x_n$ be the rising sorted sequence of numbers. The $ \sum_{k=1}^n |x_k-a|$ reaches it minimum if ...
5
votes
1answer
80 views

How we can find an equivalent of $\sum_{i=0}^n2^{2^i} $

How can we find an equivalent of the following sum: $$\sum_{i=0}^n2^{2^i}$$ Because using integrals I have been recently able to determine equivalences for a lot of sums using Riemann's theorem (sum ...
1
vote
0answers
39 views

How to manipulate this summation in the easiest way possible?

$$ D = \sum_{k=c}^{n}\sum_{j=0}^{k-c}[{k-c \choose j}\ln^{k-c-j}(g(x))[\ln(g) f'(x) f_c^{(j)} X_{n,k(f\rightarrow g)^c} + f_{c}^{(j)} X_{n,k(f \rightarrow g)^{c}}' + \frac{d}{dx}[f_c^{(j)}] X_{n,k(f ...
0
votes
3answers
24 views

Evaluate $\sum_{n=0}^{N-1} \exp(2 \pi \frac{n}{N} i)$

It is asked to evaluate the sum $$\sum_{n=0}^{N-1} \exp(2 \pi \frac{n}{N} i)$$ Using Euler's formula, the problem is reduced to evaluate $$ \sum_{n=0}^{N-1} \cos(2 \pi \frac{n}{N}) \text{ and ...
0
votes
2answers
65 views

Having trouble understanding why $\sum_{i=1}^ni^2= \frac{n(n+1)(2n+1)}{6}$ [duplicate]

So I understand $\sum\limits_{i=1}^{n}i^2 = \frac{n(n+1)(2n+1)}{6}$ but I'm not sure how to come to that conclusion. Having trouble understanding
0
votes
0answers
49 views

Multiple sums or products in wolfram-alpha

How can I compute something like $$\prod_i^n \prod_j^m ij$$ in wolframalpha? (for finite n and m) I have tried a great number of combinations that have only resulted in failure.
2
votes
2answers
44 views

Proof of factorial inequality concerning fractions

I'm having trouble with a proof, with the case $n>2$. THEOREM: For every natural number $n∈N$ where $n≠2$, $∑_{i=1}^ni≤n!$ Let us simplify the statement. ...
0
votes
1answer
29 views

Convergence of the series - best criertion

What will be the best criterion to use to investigate convergence of the series (i do not need step by step explaination) $$\sum_{n=1}^\infty \frac{e^{\frac{1}{n}} }{n^{e}}$$
1
vote
0answers
60 views

double sum with a binary variable of three elements

I have a binary variable $\ v(s,c,h)\ $which takes value 1 if subject $\ s\ $is taught in classroom $\ c\ $in time slot $\ h\ $ and 0 otherwise. I have a question about a type of constraint that I ...
2
votes
0answers
39 views

Einstein summation convention

I am not sure how to expand the following expression with regard to the Einstein summation convention. More specifically, I have: \begin{equation} a_{ij} = b_{i, j} + b_{j, i} + c_{ij, kk} \\ c_{ij} = ...
1
vote
1answer
46 views

Partitioning a set to get a sum

I have a set of numbers: 2,2,4,4,4,4,4,4,6,6,6,6,6,6 I want to enumerate the possible ways to partition this set into 4 groups, each of which sum to 16. How can I approach this short of brute force? ...
3
votes
0answers
46 views

Closed form expression for a sum

I want to calculate a sum of the form $$\sum_{k=0}^m \frac{\Gamma[m+1+\alpha-k]^2}{\Gamma[m+1-k]^2}\frac{\Gamma[x+k]}{\Gamma[x]k!}$$ where $m>0$ and belongs to integers and $\alpha$ takes half ...
1
vote
1answer
59 views

Derivative of a summation function in order to minimize the function

I'm asked to minimize this function $$f\left(x\right)= \sum_{k=1}^K \left(g\left(w\left(k\right)+\alpha\right)-t\left(k\right)\right)^2$$ with respect only to $\alpha$. Function ...
0
votes
0answers
22 views

Double summation probability [duplicate]

We define $X_i = \mathbb{1}_{\{Z_i=0,Z_{i-1}=1\}} = \mathbb{1}_{\{Z_i=0\}}\mathbb{1}_{\{Z_{i-1}=1\}}$ ($\mathbb{1}_E$ is the indicator of $E$), so that $U_n=\sum_{i=2}^n X_i$, and know that ...
1
vote
2answers
44 views

How to solve the following recurrence

I know others have already posted about this recurrence $T(n) = 2T(n/2) + n\lg n$ on the following these two posts: post1 and post2 However, the style in which they have solved them, is not one with ...
0
votes
2answers
28 views

Discrete Math-Computing Summations

So I'm asked to compute a summation with an upper limit $k = 20$ and lower limit $k=1$, where: $B_k= 0$ when $k=1$, and $B_k = \dfrac{1}{(k^2-1)}$ , for $k>1$. I was wondering if there is a ...
0
votes
1answer
54 views

Prove this sum is always zero

I encountered this problem in the context of singular homology, trying to prove that the boundary map is always zero. How do you show that the following sum, for all $n\in\mathbb{N}$, is always zero? ...
0
votes
2answers
28 views

Trying to find a formula for the following algorithm

I am trying to make a formula for the following algorithm as a function of n, building up my answer using summations. The algorithm is: ...
1
vote
4answers
21 views

Selection Sort Summation Simplification

I am trying to simplify the summation for selection sort. Starting out with: $$\sum_{i=0}^{n-1}\sum_{j=i+1}^{n-1}1$$ I am able to get: $$\sum_{i=0}^{n-1}n-i-1$$ However, I don't understand how to ...
1
vote
1answer
39 views

Summation of 1/n^2 using Fourier series on different intervals

I have been going through my notes on complex Fourier series and came across the following anomaly which I hope someone can help me with. I calculated the complex Fourier series for the function ...
4
votes
2answers
57 views

Evaluate $\sum_{n=1}^\infty \frac{n}{2^n}$.

Evaluate $$\sum_{n=1}^\infty \frac{n}{2^n}$$ My Work: $$\sum_{n=1}^\infty \frac{n}{2^n} = \sum_{n=1}^\infty n \left(\frac{1}{2}\right)^n$$ If we denote $f(x) = \sum_{n=1}^\infty nx^n$ then we ...
1
vote
0answers
19 views

How to sum random variables

Let $Z_t = \psi_t |\lambda Z_{(t-1)} + (1-\lambda)\epsilon_t |$ be a random variable where $\epsilon~N(0,1)$ is a Gaussian distributed number, $Z_0 = z_0$ and $\psi \in [-1,1]$ a random variable, ...
0
votes
0answers
42 views

How would I put these recurrence relation terms into a summation?

I was given these terms as part of a recurrence relation and I need to put it into a summation in order to solve it. $T(n)=2^{k}T\left(\dfrac{n}{2^{k}}\right) + 2^{k-1}T\left(\dfrac{n}{2^{k-1}}\right) ...
0
votes
1answer
28 views

Is there a formula for the summation of this form?

I am doing recurrence relations and I have done some work to get the summation $$\sum\limits_{i=0}^{k-1}16^{i}\left(\dfrac{n}{4^i}\right)^2.$$ I know that there is a formula if the summation was just ...
0
votes
1answer
57 views

Manipulation of summations

this question branches off another question that can be seen here Now we begin be taking a look at the following expressions: $$ \sum_{k=1}^{n-l} \sum_{j-0}^m \frac{\ln(g)^{m-j}}{g^k} ...
1
vote
0answers
18 views

Continued Fraction summation representation

I have a rational fraction of the form: $$s=\frac{p_0+p_1x+p_2x^2+\cdots+p_Mx^M}{1+q_1x+q_2x^2+\cdots+q_Mx^M} $$ The paper I am reading converts this to the form: $$s = ...
1
vote
1answer
29 views

Methods for Improving Convergence of a sequence of Partial Sums

I have the following sum: $$\zeta(3)+\frac1{4}=\sum_{k=0}^{\infty}\frac{2k^2+7k+7}{(k+1)^3(k+2)(k+3)}$$ Are there any methods that I can use to speed up the convergence of the sequence generated by ...
1
vote
2answers
48 views

finding the sum function of $\sum_ {n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1}$

finding the sum function of $\sum_ {n=1}^{\infty} \frac{n-2}{(n-1)!} z^{n+1}$ So far i've substituted n-1 for m which gives me the following form: $\sum_ {m=0}^{\infty} \frac{m-1}{(m)!} z^{m+2}$. ...