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

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2
votes
1answer
39 views

A series whose terms are the products of terms of a geometric and a power series

Consider this summation $$ \sum_{i=1}^{\infty}\frac{1}{i^ab^i} $$ where $a$ and $b$ are greater than $1$ It can be upper bounded by the geometric series ...
0
votes
2answers
53 views

Integral To Summation Problem

$\int x^n e^{cx}\; \mathrm{d}x = \frac{1}{c} x^n e^{cx} - \frac{n}{c}\int x^{n-1} e^{cx} \mathrm{d}x = \left( \frac{\partial}{\partial c} \right)^n \frac{e^{cx}}{c} = e^{cx}\sum_{i=0}^n ...
1
vote
2answers
51 views

The meaning of a Series VS Integral

This may look as a stupid question but it will really help me a lot in understanding some things. Suppose I take the simple function $f(x) = x$. Now I will evaluate two different operations: ...
3
votes
1answer
68 views

Any simpler form for $ \frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{k}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$

Is there any simpler form for the following expression: $$ \frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{k}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$$ Because I have to compute this ...
1
vote
1answer
42 views

Proof By Induction With Integration Problem

I am required to prove this formula by induction$$ \int x^k e^{\lambda x} = \frac{(-1)^{k+1}k!}{\lambda^{k+1}} + \sum_{i=0}^k \frac{(-1)^i k^\underline{i}}{\lambda^{i+1}}x^{k-i}e^{\lambda x}$$ where ...
0
votes
1answer
57 views

How do I write $F_1+F_3+F_5+\ldots+F_{2n-1}$ in summation notation?

How do I write $F_1+F_3+F_5+\ldots+F_{2n-1}$ in summation notation? $F_i$ represents the Fibonacci sequence. I can't figure out how to write this in summation notation. Clear steps would be ...
2
votes
1answer
216 views
+100

Help with proving a statement based on riemann sums?

Suppose we have a reimmen sum with no removed partitions which I call the "total sum". $$\lim_{n\to\infty}\sum_{i=1}^{n}f\left(a+\left(\frac{b-a}{n}\right)i\right)\left(\frac{b-a}{n}\right)$$ And ...
0
votes
1answer
36 views

Maximize equation with summation.

Given: $$l(\lambda) = -\lambda n + log(\lambda) \sum_{i=1}^n x_i - \sum_{i=1}^n log(x_i!)$$ Verify that this function $l(\lambda)$ is maximized by $\bar{x} = \sum_{i=1}^n x_i$ Taking the ...
1
vote
0answers
49 views

Closed form of sum of rounded number

Let, there is two variable $N$ and $D$. Now, I want to find the closed form of the following sum: $$S_n = \sum_{k=1}^N \lfloor{\frac{k}{D}+\frac{1}{2}}\rfloor$$ Its easy to find the closed form by ...
0
votes
1answer
41 views

Problem with summations for sorting algorithm?

According to this lecture http://web.stanford.edu/class/archive/cs/cs161/cs161.1138/lectures/10/Small10.pdf, slide 26, the expected number of comparisons done by quicksort is smaller or equal to ...
0
votes
2answers
34 views

Proof of sum-of-i formula [duplicate]

What is the proof of $$\sum\limits_{k=1}^n(2k-1)=n^2$$ I understand that it derives from $\sum\limits_{k=1}^nk=\frac{n(n+1)}{2}$ but I miss how to relate that proof for this one.
3
votes
1answer
30 views

Simplify double summation

I need to simplify this to find the Big-Oh but I am not very familiar with double summations (second element is my attempt but it might be wrong): $$\sum\limits_{i=1}^n \sum\limits_{j=2}^{2n+1}1 = ...
0
votes
0answers
30 views

How to write the mathematic formulation of a pairwise sum?

In the normal summation, we can do this: And that would sum a list X as such in Python: ...
1
vote
1answer
41 views

Relation between two sequences or summations

Let us define two sequences \begin{equation} G_{n}=\sum_{i=1}^n a^{-i}t_{i-1} \end{equation} and \begin{equation} g_{n}=\sum_{i=1}^n t_{i-1} \end{equation} where $a$ is an integer and $t_n$ is an ...
2
votes
1answer
38 views

Abel summability of arithmetic means of a sequence [closed]

Let $\displaystyle\lim_{x\to1^-}(1-x)\sum_{n=0}^{\infty}s_nx^n=s$ for $|x|<1$, i.e. the sequence $(s_n)$ be Abel summable to $s.$ How to prove the sequence of arithmetic means $\displaystyle ...
1
vote
4answers
101 views

Any simpler expression for$\frac{\sum_{k=2}^{n-2}{k\big(\sum_{i=0}^{n-2}\frac{(-1)^i}{i!}\big)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$

Is there any simpler form for the following expression: $$ \frac{\sum_{k=2}^{n-2}{k\left(\sum_{i=0}^{n-2}\frac{(-1)^i}{i!}\right)}}{n\sum_{i=0}^{n}\frac{(-1)^i}{i!}}$$ Because I have to compute this ...
1
vote
1answer
42 views

Prove that the two powers are equal

Prove that: $$\dfrac{1}{2^{180}a^{360}}\dfrac{(a^{720}-1)(a^2-1)}{a^{2}+1} = \dfrac{\left(1+\dfrac{\sqrt{3}}{2}\right)^{180} - \left(1-\dfrac{\sqrt{3}}{2}\right)^{180}}{\sqrt{3}}$$ where: $$a = ...
1
vote
1answer
47 views

Determine the optimal number and location of the plants and pipelines in the 7 cities.

There are 7 cities, up to 4 plants to be made in them and up to 18 pipelines to be made connecting them. Determine the optimal amount of plants and pipelines to be made, the optimal locations of the ...
1
vote
0answers
45 views

Develop a model for determining the optimal production schedule in a manufacturing facility

I have to formulate (linearly) the following problem mathematically: What I tried: 1. Variables Let $x_{ijk} = 1$ if, in month k, product i should be made in production line j, where ...
1
vote
5answers
41 views

Sum of the Powers of $2$

Suppose I have a sequence consisting of the first, say, $8$ consecutive powers of $2$ also including $1$: $1,2,4,8,16,32,64,128$. Why is it that for example, $1 + 2 + 4 = 7$ is $1$ less than the next ...
0
votes
0answers
46 views

How to show the following series diverge or converge if we assume $\lim\limits_{k\rightarrow \infty}a_k=0$?

For $\sum_{k=0}^{\infty} {a_k^2}$, $\sum_{k=0}^{\infty} {\sqrt{|a_k|}}$ and $\sum_{k=0}^{\infty} {\frac{a_k}{k}}$, if i understand the question correctly. I think we can give both convergent and ...
1
vote
1answer
24 views

Compute $\sum_{i=0}^{j} {\frac{(-1)^i}{i!}(j+1)!(1+i)}$

I am trying to figure out how to evaluate the following sum: $$\sum_{i=0}^{j} {\frac{(-1)^i}{i!}(j+1)!(1+i)}$$ I have no idea how to proceed with this, but I really need to use it because it is part ...
2
votes
3answers
60 views

Is $n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! \bigg[\sum_{i=0}^{n-2}{\frac{(-1)^i}{i!}}+…+\sum_{i=0}^{2}{\frac{(-1)^i}{i!}}\bigg]=(n-1)!$ true?

I am in the middle of doing a problem and has this sort of expression. I have a feeling that the following equality holds: $$n! \sum_{i=0}^n{\frac{(-1)^i}{i!}}- (n-1)! ...
1
vote
1answer
36 views

Is $\sum_{n=0}^\infty (a \cdot r^n)$ equivalent to $\lim_{n \to \infty}\sum_{k=0}^n (a \cdot r^k)$?

In other words, when writing down an infinite sum, are we always implying that it's actually the limit of that series as the number of terms approaches infinity, or is there some subtle difference?
0
votes
0answers
18 views

Linearizing Summation in Replica Trick (Using "well-known properties of Gaussian Integrals)

I am reading Dotsenko's Introduction to the Replica Theory of Disordered Statistical Systems and Parisi/Mezard's Spin Glass Theory and Beyond. In both books, we have: $$ ...
0
votes
1answer
13 views

Separating variables from inequality. [duplicate]

Let, $$S_n = \sum_{1\leq j<k+j\leq n}^{} 1/k$$ In the book concrete mathematics, The next line shows that, $$S_n = \sum_{k=1}^n \sum_{j=1}^{n-k} 1/k$$ I am getting very hard times understanding ...
0
votes
1answer
41 views

Simplify Summation of combination [duplicate]

how can I simplify this to a phrase without a sigma?! $$ {1\over r+1}{2r \choose r} + \sum_{i=1}^r \left( {i+1\over r+1} {2r-i \choose r-i}{s+i-2 \choose i} \right) $$ thanks!
5
votes
2answers
102 views

proof - Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n=3$

Show that $1! +2! +3!+\cdots+n!$ is a perfect power if and only if $n =3$ For $n=3$, $1!+2!+3!=9=3^2$. I also feel that the word 'power' makes it a whole lot hard to prove. How do we prove this? ...
0
votes
1answer
25 views

simplifying summations

From knowing $7c=\sum_{i=1}^{50-c}k_i$ and $c\choose 2 $=$ \sum_{i=1}^{50-c}$ $k_i\choose 2 $ how can I get to $\sum_{i=1}^{50-c}(k_i-\mu)^2=(50-c)\mu^2-14c\mu+c^2+6c$ for some arbitrary ...
2
votes
2answers
80 views

Sum of all Products on Catalan numbers

how can I simplify this? let: $$ C_n = {{2n \choose n}\over n+1} $$ find: $$ \sum_{P_1 + P_2 + ... + P_k = r} \left(\prod_{j = 1}^k C_{P_j}\right) $$ thanks!
-1
votes
1answer
38 views

Convergent series of fractions

Is $\sum_{n=1}^\infty \frac{a}{n}$ guaranteed to converge for any value of a? Or does it simply go to infinity? I'm most interested in a = 1, but it is also interesting to see whether there is a ...
1
vote
2answers
38 views

Computer Vision Models 4.3 - Derivative of Summation

I am reading through the Computer Vision: Models, Learning, and Inference book to get an understanding of computer vision. The author describes the high-level steps taken to arrive at one of the ...
2
votes
2answers
38 views

Prove expression is positive

Let $\sum_i w_i=1$ and $w_i, x_i \in \mathbb{R}$, show that $$\sum_i w_i x_i^4-\sum_i w_i x_i \sum_i w_i x_i^3\geq 0. $$ I can show that $\sum_i w_i x^2 -\sum_i w_i x_i \sum_i w_i x_i \geq 0$ by ...
2
votes
4answers
91 views

Numerically efficient summation of shrinking series $\sum_1^n 2/(2k - 1)$

Let $$G_{2n} := G_{2n+1} := -\gamma -\log 2 + \sum_{k=1}^n \frac{2}{2k-1}$$ where $\gamma$ is Euler's constant. (We also have $G_1 := -\gamma - \log 2$.) Thus we have $$G_{2n+2} = G_{2n} + ...
0
votes
1answer
36 views

Find $\sum_{k=0}^6cot({\pi\over 21} + {k\pi\over 7})$

Find $$\sum_{k=0}^6cot({\pi\over 21} + {k\pi\over 7})$$ I don't know how to do this at all. I initially started using complex numbers but couldn't get it. Please give a simple solution.Thanks.
0
votes
1answer
23 views

solving summation problem when the lower limit is a negative integer

How can I do this problem? I am unsure as to whether I can modify the range of the sum so that i = 1, and n = n+13. Help! $$\sum_{i=-12}^{n} 2i^{2}$$
0
votes
2answers
23 views

$\sum_{x \in \mathbb{Z}^n} e^{-2x^Tx}= \left(\sum_{y=-\infty}^{\infty} e^{-2y^2}\right)^n$ is true?

Is the following true? $$\sum_{x \in \mathbb{Z}^n} e^{-2x^Tx}= \left(\sum_{y=-\infty}^{\infty} e^{-2y^2}\right)^n\;?$$
0
votes
4answers
30 views

Show that $0<\alpha p+ \beta(1-p)<1$ for $\alpha, \beta, p$ $\in (0,1)$

Consider $0<\alpha<1$, $0<\beta<1$, $0<p<1$. Are these sufficient conditions for having $$ 0<\alpha p+ \beta(1-p)<1 $$ ? Hint for the proof?
0
votes
1answer
26 views

Summation of terms of an exponential progression.

I was recently considering a progression where each term in the sequence is the previous term raised to a common exponent. To elucidate: $$S_{E.P}(a,m)=a,a^m,{(a^m)}^m,({(a^m)}^m)^m \cdot \cdot ...
0
votes
1answer
35 views

Differentiating $\sum_{k=1}^{n}k$

I'm trying to prove the following using differentiation. $$\sum_{k=1}^{n} k=\frac{n^2+n}{2}$$ Looking all over the place, I see no rules for deriving such sums. If I use the limit function to find ...
0
votes
1answer
19 views

Proof of the identity $ \sum_{i=n}^{N} \sum_{m=i}^{N} (-1)^{m-i} \binom{m}{i} x_m= \sum_{i=n}^{N} (-1)^{i-n} \binom{i-1}{n-1} x_i $

I want to show the identity $$ \sum_{i=n}^{N} \sum_{m=i}^{N} (-1)^{m-i} \binom{m}{i} x_m= \sum_{i=n}^{N} (-1)^{i-n} \binom{i-1}{n-1} x_i, $$ where $x_1, \ldots, x_n \in \mathbb{R}$. By first ...
0
votes
4answers
31 views

Show that $\sum_{i=1}^n a_i p_i=1$ if and only if $p_i=1$ when $0<a_i<1$, $\sum_{i=1}^n a_i=1$, $0\leq p_i\leq 1$

Consider $n$ real numbers $0<a_i<1$ such that $\sum_{i=1}^n a_i=1$ Consider other $n$ real numbers $0\leq p_i\leq1$. Could you help me to show that $\sum_{i=1}^n a_i p_i=1$ if and only if ...
1
vote
2answers
19 views

Summation of stopping point

I'm learning summation and I need help with the following sum: $\sum\limits_{i=0}^{n-2} n$ What I thoughts is, since $n$ don't change, my sum will be $S_n$ = $n_1$ + $n_2$ + $n_3$ + ... + $n_{n-2}$ ...
7
votes
1answer
115 views

A query about Poisson summation and matrices

I am trying to reproduce a proof I saw a while ago and it requires an equality similar to the following. Here $M$ is a real-valued positive semi-definite matrix. $$\sum_{x \in \mathbb{Z}^n}e^{-x^TMx} ...
5
votes
3answers
96 views

Is $\sum_{x\in\mathbb{Z}^n} e^{-x^Tx} < 2^n$?

Is it possible to find an upper bound (or even an exact value) for $$\sum_{x\in\mathbb{Z}^n} e^{-x^Tx}\;?$$ In particular, is this sum less then $2^n$? Approximate numerical answers: For ...
2
votes
1answer
35 views

Factoring inequalities on Double Summation (Donald Knuth's Concrete Mathematics)

If you have the Concrete mathematics book please refer to page 40 and 41. So how come this given sum $$ \sum_{1 \le j < k + j \le n} \frac{1}{k} $$ becomes $$ \sum_{1\le k \le n}\sum_{1\le j ...
2
votes
3answers
133 views

An upper bound for $\sum_{n=1}^{\infty}e^{-n^2}$

I am trying to find a good upper bound for $$\sum_{n=1}^{\infty}e^{-n^2}\approx 0.3863186024.$$ I know that $$\int_{x=0}^{\infty}e^{-x^2}\;dx=\frac{\sqrt{\pi}}2 \approx 0.8862269255.$$ Is it ...
1
vote
1answer
41 views

Choose which plants to build to maximise profit

What I tried: Let $y_i = 1$ if plant $i$ is to be constructed and $0$ otherwise Let $c_{ij}$ be transportation cost per-unit for whatever the plants produce delivered from plant $i$ to ...
0
votes
2answers
85 views

Maximise population coverage subject to budget constraint

Let $t_i =$ $1$ if transmitter i is to be constructed and $0$ otherwise, $c_j =$ $1$ if community j is covered and $0$ otherwise. Obj func: Max $$z = [10, 15, ..., 10] \cdot c$$ s.t. ...
0
votes
1answer
21 views

Summation of$ n$ sub of $i - 1$?

this is a simple summation but i am not sure how to get from the left side to the right side. Any help would be appreciated. thank you All I know is that if it is only $n$ sub of $I$, then it will be ...