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

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0
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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
32 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 ...
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vote
2answers
20 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
128 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
36 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
134 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
44 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
88 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 ...
1
vote
2answers
49 views

Convergence or divergence of $\sum_{n=1}^\infty (-1)^{n+1}(2+(-1)^n)/n$? [duplicate]

Is $$\sum_{n=1}^\infty (-1)^{n+1}{(2+(-1)^n)\over n}$$ convergent or divergent?
1
vote
1answer
39 views

How can I calculate $\sum_{k=1}^{n-1}\binom{n-1}{n-k}$?

I would like to know if I can calculate a closed expression for $$\sum_{k=1}^{n-1}\binom{n-1}{n-k}$$ This sum is equals to: $$1+(n-1)+(n-1)(n-2)+(n-1)(n-2)(n-3)+\ldots+(n-1)(n-2)/2+(n-1)$$
3
votes
1answer
42 views

What is condition that the sum of $n$ complex numbers eaquals their product

Let $n\geq2$ and let $\{z_1,\dots,z_n\}$ be a set of complex numbers. Is there a condition on the $z_i$'s such that $$\sum_{i=1}^n z_i=\prod_{i=1}^n z_i$$ is identically true? For $n=2$ the ...
2
votes
1answer
40 views

The sum of k times the kth power of a is given analytically by?

I was wondering how would someone derive (Not prove) the result in terms of n and a of the following sum: $$\sum_{k=1}^n ka^k$$ My question basically is, given that summation how would you tackle ...
1
vote
0answers
24 views

Binomial expansion for harmonic numbers

It is well known that harmonic numbers in general cannot be expressed through elementary functions. I am interested in following sums: \begin{equation} {\mathcal S}_\theta(n):=\sum\limits_{p=0}^n ...
0
votes
1answer
60 views

if $f(n) \to g(n)$ then $\frac{\sum f(n)}{\sum g(n)} \to 1$?

Say we have some function $f(n)$ that behaves like $g(n)$ for large $n$. It is easier to analyse in general the $g(n)$. Then it seems intuitive to say $$\frac{\sum f(n)}{s_n}\to 1$$ where $s_n= \sum ...
0
votes
0answers
78 views

Infinite series with e and pi

Please, help me to prove that $$ \sum_{n=1}^\infty \frac{\cos(\pi n-\sqrt{\pi^2n^2-9})}{n^2}=-\frac{\pi^2}{12 e^3} $$ I found this fact here https://en.wikipedia.org/wiki/List_of_representations_of_e ...
1
vote
1answer
64 views

Closed form of this expression

So at my school today, we celebrated Pi Day, and we had a customary mini-olympiad. In that olympiad there was a really interesting question. There is a sequence: $\lbrace 1\rbrace, \lbrace 4, ...
4
votes
0answers
82 views

Proof of Sophomore's Dream using Contour Integration

Sophomore's dream is a relatively common identity, that states $$ \int _0^1 x^{-x} dx = \sum_{n = 1}^\infty n^{-n}$$ The common proof is found using the series expansion for $ e^{- x \log x} $ and ...
-1
votes
1answer
32 views

Knowing the time number of times print function is called here and how can I express this sudo code in a summation form

for i from 1 to n do for j from 2 to n*n do print ‘Q’ end do print newline end do For the double for loop, which is a pseudocode I am ...
-1
votes
3answers
28 views

Summation using sum rules [duplicate]

I have this question I am currently trying to figure out: $$\sum_{i=-12}^n 2i^2$$ How can I solve this question please? Thanks for your help.
2
votes
1answer
35 views

Quasi-shuffle identity

I tried to evaluate the following sum: $$S=\left( \sum _{ { m }_{ 1 }=1 }^{ n }{ \frac { 1 }{ { m }_{ 1 } } } \right) \left( \sum _{ { m }_{ 2 }=1 }^{ n }{ \frac { 1 }{ { m }_{ 2 } } } \right) ...
0
votes
2answers
49 views

Evaluation of sum of digamma function.

while solving a summation problem I got stuck at this: $$\sum _{ n={ 2 }^{ k-1 }+1 }^{ { 2 }^{ k } }{ \left( \psi \left( 2n \right) \right) } $$ The limits of this summation are weird and ...
5
votes
1answer
75 views

Reference request for an identity involving binomial coefficients

The identity is $$\sum_{i\ge 0}(-1)^i \binom{\frac{s^k + s^{-k} - 10}{4}}{i}\binom{\frac{s^k + s^{-k} - 10}{4}+4}{\frac{gs^k + 2g^{-1}s^{-k} - 4}{8}-i}= 0$$ where $k\gt 1$ , $s = 3+2\sqrt{2}$ ...
0
votes
0answers
50 views

Sum of Two Double Sums

Suppose $ r^{2}=4ab $. Show that over the complex number field: $$ \left( \sum_{k=0}^{l}{\sum_{m=0}^{l-k}{\binom{l}{k}\binom{l-k}{m}a^{l-k-m}b^{m}(-1)^{m}r^{k-1}i^{k-1} \left( ...
3
votes
2answers
60 views

How to evaluate $\sum_{k=1} ^{n-1} \frac{\sin (k\theta)}{\sin \theta}$

How to evaluate $$\sum_{k=1} ^{n-1} \frac{\sin (k\theta)}{\sin \theta}$$ Any help ? I tried to use difference method. But I'm not getting there.
4
votes
2answers
110 views

Find: $\zeta \left( 3,1,1,1 \right)$

While solving a summation, I came across this: $$\zeta \left( 3,1,1,1 \right)=? $$ I'm new to multiple zeta values. That's why I couldn't find this. So my question is does a closed form exist ...
12
votes
4answers
569 views

Is Sigma $\Sigma$ a mathematical way of doing a for loop?

I've been a programmer for ten years, and once upon a time I was pretty good at math. Those days are long gone. I'm taking some online classes and now I find myself needing to remember the math I ...
5
votes
4answers
132 views

Finding the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$

I am prepping for my mid semester exam, and came across with the following question: Find the closed form for the sum $\sum_{j=k}^n (-1)^{j+k}\binom{n}{j}\binom{j}{k}$, using the assumption that ...
4
votes
0answers
81 views

Finite Messy Trigonometric Sum

Show the following result:$$\sum_{m=1}^{99}{\frac{\sin{\left(\frac{17 m \pi}{100}\right)} \sin{\left(\frac{39 m \pi}{100}\right)}}{1+\cos{\left( \frac{m\pi}{100} \right) }}}=1037$$ The source of this ...
0
votes
1answer
39 views

Approximation to $\displaystyle\sum_{n=0}^{x}n^k$ with integrals

Taking a quick look at a graph, it should be fairly obvious that $$\int_0^xn^kdn<\sum_{n=0}^xn^k<\int_0^{x+1}n^kdn$$ If $k>1$ and $x=1,2,3,\dots$ I was wondering if there existed tighter ...
2
votes
2answers
52 views

Finding the sum of a series, don't understand how they get from one step to the next

Can some one please explain how they get from the red boxed step to the blue boxed one?
12
votes
5answers
1k views

What is the sum of the reciprocal of all of the factors of a number?

Suppose I have some operation $f(n)$ that is given as $$f(n)=\sum_{k\ge1}\frac1{a_k}$$ Where $a_k$ is the $k$th factor of $n$. For example, ...
1
vote
1answer
65 views

How can I find this sum?

I'm doing some examples related to convolution (digital signal processing). I post my problem here because it is actually mathematics problem. I have to calculate this sum: $$\sum_{k\ = \ n-5}^{n+5} ...
1
vote
1answer
47 views

sum of roots of unity multiplied by k+1

I'm considering the following sum: $$\sum\limits_{k=0}^n (k+1)\epsilon^k,$$ where $\epsilon=e^{\frac{2\pi i}{n}}$. I write the sum as $$\frac{\rm d}{{\rm d}\epsilon}\sum\limits_{k=0}^n ...
0
votes
0answers
30 views

Evaluating a series practically

I am trying to figure out a practical approach to finding the finite result of an infinite sum - preferably with my HP-50G calculator. For example, if given the following function: $$i_f(t) = 1.25 + ...
1
vote
3answers
64 views

$3$-adic expansion of $- \frac{9}{16}$

I get the $3$-adic expansion to be $1+1 \cdot 3+2 \cdot 3^2 +2 \cdot 3^3 + 0 \cdot 3^4+\cdots$. I'm trying to work out a pattern of the coefficients and think it is $1, 1, 2, 2, 0, 0, 1, 1, 2, 2, 0, ...
1
vote
1answer
22 views

Limit of products in $x_n = 1-An^{-\alpha}$ and their summation

Suppose that we have $A >0, \alpha >0$, and for each $n$, define $x_n = 1-An^{-\alpha}$ such that for large $n$ we have $x_n \in (0,1)$. Also, define the product sequence, $y_n = \prod_{i=0}^n ...
2
votes
2answers
52 views

Alternate method to prove this series in a better way

Prove that $\frac{1.2 + 2.3 + 3.4 + .....+ n(n + 1)}{n(n + 3)} \ge \frac{n + 1}{4}$ for $n\ge1$ My attempt : Breaking the series into two different series $$ S_1 = \sum_{i = 0}^n i^2 = \frac{n(n + ...
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0answers
33 views

Turn summation into matrix product

This problem is a tad complex, I'll try to explain. We have a function $S$ which is defined on $\mathbb R^{m \times d}$ matrics. We are given the formula $\frac{\partial s}{\partial w_{1_i}} = ...
1
vote
2answers
345 views

I am stuck on proving $\frac1{2!}+\frac2{3!}+\dots+\frac{n}{(n+1)!}=1-\frac1{(n+1)!}$ by induction, could anyone check my work?

I will skip the Base Case step. This is the questions. Use mathematical induction to prove that$$\frac{1}{2!}+\frac{2}{3!}+\cdots+\frac{n}{(n+1)!}=1-\frac{1}{(n+1)!}$$for all integers $n\ge 1$. ...
0
votes
0answers
29 views

Renewal Processes (Rolling a Fair Die)

What is the expected number of rolls until you see a 6 immediately followed by a 2? Let $D_i$ = number observed on the $i^{th}$ roll Assume $T_0=0$ Now let $T_1$ = min{$i\ge2$:$D_{i-1}=6$, ...
2
votes
0answers
24 views

Finding a generating function for a sequence with two recurrence equations

The sequence $a_{n}$ is defined as follows: $a_{0}$ = 0 , $a_{1}$ = 1 $a_{2n} = a_{n}$ $a_{2n+1} = a_{n} + a_{n+1}$ let the generating function $F(x)$ be defined as $F(x) = \sum_{n=1}^{\infty} ...
1
vote
1answer
86 views

Summation of square root sine

Is it possible to solve this? $$y = \sum_{N=1}^{x}\sqrt{\sin N}$$ I know it is possible to solve $$y = \sum_{N=1}^{x}{\sin N}$$ by expressing sinN as its exponential before doing geometric ...
2
votes
1answer
38 views

Mistake in proof for sequence of cubes

I know there are thousands of proofs for this to have a look at, but I started one myself in a slightly different way than what is easily found when googling. To me, the proof seems like it should ...
0
votes
1answer
34 views

Need assistance deriving a complicated function

I'm trying to prove a formula regarding the derivative of a somewhat complicated function. $w \in \mathbb R^M$ is our variable, we have a set of $M$ functions $\phi_j: \mathbb R \to \mathbb R$ a ...
0
votes
2answers
38 views

Find the closed form for the double sum $\sum_{i=1}^n \sum_{j=i}^n 2j$

This is what i get: $$n^3 + n^2 - n(n+1)(2n+1)/6 + n(n+1)/2$$ When I simplify : I get : $(1/3)n(2n^2+3n+1)$ Is anyone else getting the same result.