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

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8
votes
2answers
93 views

Rough bound for sum $\binom{3n}{0}+\binom{3n}{1}+\cdots+\binom{3n}{n-1}$

Is it true that $$\frac{\dbinom{3n}{0}+\dbinom{3n}{1}+\cdots+\dbinom{3n}{n-1}}{2^{3n}}<\frac13$$ for all positive integers $n$? I've plotted the first few values of $n$ and noticed that the ...
2
votes
3answers
31 views

$\sum^6_{i=1}(x_i-\bar{x})^2$ as $\sum^6_{i=1}x_i^2 - 6\bar{x}^2$ what rules where applied?

consider the set $X = \{20, 30, 40, 50, 60, 70\}$ and the mean $\bar{x} = 45$ then $\sum^6_{i=1}(x_i-\bar{x})^2 = 1750 = \sum^6_{i=1}x_i^2 - 6\bar{x}^2$. How would I transform the first term by hand ...
1
vote
2answers
48 views

Ratio of two summations

I devised this question based on recent (and not-so-recent) MSE questions on summations. Evaluate ...
0
votes
1answer
25 views

time complexity of an algorithm

Hi all i'm trying to predict/calculate the time complexity of an algorithm but i'm having some difficulties with the summations the algorithm: ...
2
votes
0answers
21 views

evaluation summation Erlang distribution

I have to calculate a renewal function: $H(t)$. In which: $$ H(t)=\sum_{n=1}^\infty \left(1-e^{-\lambda t}\sum_{i=0}^{2n-1}\frac{(\lambda t)^i}{i!}\right) $$ I think it can be solved by switching ...
1
vote
2answers
35 views

Which is greater ? Sum of odd power terms or even power terms in the exponential Taylor series?

I came across this question, in a book. Define $f(x) = \sum_{n=0}^{\infty} \frac{x^{2n+1}}{{(2n+1)}!} $ and $ g(x) = \sum_{n=0}^{\infty} \frac{x^{2n}}{{(2n)}!} $, where x is a real number. Then, ...
0
votes
0answers
42 views

sum of zeta function [duplicate]

how do I solve this question? $$\sum_{k\geq2} ( \zeta(k)-1) $$ I know that $\zeta(2)$ is $\frac{\pi^2}{6}$ and $\zeta(k)$ can be represented as $$\sum_{j\geq1} \frac{1}{j^k}$$ Thanks in advance!
0
votes
1answer
27 views

On a trigonometric summation.

I have a function defined as: $$C_n(w) = \frac{1}{n} \sum_{j = 0}^{n-1}\sum_{k=0}^j (a_k \cos(kw) + b_k \sin(kw)) $$ Now it is stated that: $$C_n (0) = \frac{a_0}{2} + \frac{1}{n} \sum_{r = 2}^n ...
1
vote
4answers
97 views

Calculate the sum $\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$

$$\sum_{n=3}^{\infty}\frac{4n-3}{n^3-4n}$$ I think it is related to power series, because it is the topic, but I have no idea how to get there. Could you give a hint?
13
votes
4answers
305 views

How to prove $ \sum_{k=0}^n \frac{(-1)^{n+k}{n+k\choose n-k}}{2k+1}=\frac{-2\cos\left(\frac{2(n-1)\pi}{3}\right)}{2n+1}$

How to prove $$\sum_{k=0}^n \binom{n+k}{n-k}\frac{(-1)^{n+k}}{2k+1}=-\frac{2}{2n+1}\,\cos\left(\frac{2(n-1)\pi}{3}\right)\;\text{?}$$ I have a proof by induction for it, but it isn't simple! I want ...
0
votes
1answer
87 views

About a sum involving factorials.

I would like to know if there is a closed form of $$\sum_{k=0}^{n}\frac{4^{k}}{\left(2k\right)!\left(n-k\right)!^{2}}.$$ Wolfram gives a strange closed form and, i.e., ...
1
vote
2answers
103 views

If a vector v is an eigenvector of both matrices A and B, is V an eigenvector of A+B? [closed]

If so, is there a proof for this? I have been stuck trying to validate the statement and would love some insight.
6
votes
2answers
104 views

Changing-sided dice probability problem.

Suppose you roll a fair $6$-sided dice, and that the number you roll is $m$. If $m=1$, stop. Otherwise, roll an $m$-sided dice. The number you roll is $n$. If $n=1$, stop. Otherwise roll an $n$-sided ...
0
votes
1answer
23 views

Question about double summation notation.

Just started learning about double integrals literally $10$ minutes ago. I have a fairly good grip on the Riemann integral and so far it seems very similar, but we are just working with volumes ...
0
votes
1answer
25 views

Combinatorics-Summation doubt in the proof of the expectation of the Hypergeometric distribution.

The proof starts considering this equality: $(d/dx (1+x)^A)(1+x)^B = A(1+x)^{A+B-1}$ Then it keep on changing every $(1+x)^{A or B}$ for its binomial coefficient. That 's what I don't understand. If ...
16
votes
5answers
117 views

Geometrical interpretation of $(\sum_{k=1}^n k)^2=\sum_{k=1}^n k^3$

Using induction it is straight forward to show $$\left(\sum_{k=1}^n k\right)^2=\sum_{k=1}^n k^3.$$ But is there also a geometrical interpretation that "proves" this fact? By just looking at those ...
2
votes
1answer
33 views

Find the probability generating function $G(s)$ of this branching process.

Suppose that $X_n$ is size of the $n$th generation of a branching process started from a single individual, where each individual has a random number of children with probability mass function: ...
6
votes
0answers
108 views

Partial sums of falling factorials

I want to know if there exists some way, approximate or exact, to do a partial sum of falling factorials of the kind: $$\sum_{k=i}^{n}(a+k)_{h}$$ where all are constants. And I'm interested too in ...
1
vote
1answer
31 views

Looking for help in regard to Series solutions with ordinary points (ODE)

I have a question that is in regard to the final answer that one is to get when solving some ODE questions via series. I am having some confusion on what if I am doing is correct/ why it is or is not ...
2
votes
2answers
48 views

How does this manipulation of summations work?

I am reading some mathematics in which is the following algebraic manipulation. $$ \begin{align} \exp(x)\exp(y) & = \left(\sum_{n = 0}^\infty \frac{x^n}{n!}\right) \left(\sum_{m = 0}^\infty ...
0
votes
0answers
43 views

Sum of powered numbers

Here is what I have achieved so far: $S(n,p)= 1^p+2^p...+n^p$ By arranging it this way: $\ \ \ \ \ \ \ \ 1^{p-1}$ $\ \ \ \ 2^{p-1} 2^{p-1}$ $3^{p-1} 3^{p-1} 3^{p-1}$ $..............$ I ...
3
votes
3answers
98 views

evaluate the sum $\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$

I'm trying to evaluate this sum $$\sum_{n=1}^{\infty}\sum_{k=n}^{\infty}\frac{1}{(n^2+n-1)(k^2+k-1)}$$ I have no idea how to deal with it. With one sum I can, with partial-fraction decomposition, ...
2
votes
2answers
67 views

Nested sum $\sum_{i<j< \cdots < k} ij \cdots k$

I am wondering if there is any known closed form for the following nested sum? : $$ \sum_{i<j<\cdots <k} ij\cdots k $$ where each $i,j,\cdots,k =1, \cdots, n$ I tried the first one: $$ ...
2
votes
2answers
53 views

Alternating Sum of Cubes [closed]

How is it possible to evaluate: $$\sum_{k=1}^n{((-1)^{n-k}\cdot k^3)}=n^3 - (n-1)^3 + (n - 2)^3 - \cdots \pm 1^3$$ The fact that there is the $\pm$ at the end makes it difficult.
2
votes
1answer
46 views

How do we derive the sum of $3^n$ and $2^n$

I know that $\quad\sum2^n = 2 (2^n-1)$ How can we derive this summation? And also how can we deduce the summation of $3^n$ from this ? I did observe this pattern : $$ \begin{align} n &= 1 ;\ ...
4
votes
2answers
57 views

Multiplication Principle and Inclusion-Exclusion: $2^n = \sum_{i = 0}^n (-1)^i \binom{n}{i} \binom{2n - 2i}{n - 2i}$

I began to compose an unnecessarily complicated answer to this question: If we had 25 people all who have 2 different balls, how would you work out how many combinations there would be if we want ...
1
vote
1answer
35 views

Understanding summations with Poisson

I'm currently doing a problem on Poisson processes and I've encountered the situation where I'm not sure why this summation is expanded as follows: And similarly I have tried expanding out the ...
3
votes
1answer
56 views

Exact value of a sum involving harmonic numbers

Could somebody tell me the exact value of this series? $$ \sum_{k=1}^{\infty} (-1)^k\frac{H_k^{(5)}}{k} $$ where $$ H_k^{(n)}=\sum_{i=1}^{k}\frac{1}{i^n} $$ Thanks!
0
votes
1answer
27 views

Factoring constant in summation

Trying to show that adding a constant c to $\sum_{k=0}^\infty a^kx_k$, where a is a constant will just add some constant k to the summation eg. $$\sum_{n=0}^\infty a^n(x_n+c)$$ $$=k + ...
2
votes
4answers
69 views

How can I simplify $1\times 2 + 2 \times 3 + .. + (n-1) \times n$ progression? [closed]

I have a progression that goes like this: $$1\times 2 + 2 \times 3 + .. + (n-1) \times n$$ Is there a way I can simplify it?
1
vote
1answer
28 views

Expanding a term with a sum

We have the following quantity: $$E\left[\left(\sum^n_{j=1} (X(t_j) - X(t_{j-1}))^2-t\right)^2\right]$$ My textbook says this can be expanded in the following way (colors are my touch) ...
1
vote
2answers
83 views

Closed form of sum with binomial

I want to find closed form of the following expression : $$\sum\limits_{k=0}^{n} \binom{n}{k}\frac{(-1)^k}{2k+1}$$ I have no idea how to do it.
0
votes
2answers
36 views

Approximate summation of flooring function

I have this summation: $\sum_{k=0}^x \lfloor{\frac{k}{c}}\rfloor$ Do you have any ideas on any general expressions that can approximate this? P.S I know I can approximate it with a Fourier ...
2
votes
1answer
46 views

Sum of the series with Stirling numbers of the first kind.

Yesterday I worked on one problem in discrete math and in the process of decision I came across this series. Try to do it with generating functions, but there is no success for me. So, what do you ...
0
votes
1answer
34 views

Summation operation for precalculus

Studying Spivak's Calculus I came across a relation I find hard to grasp. In particular, I want to understand it without using proofs by induction. So please prove or explain the following ...
1
vote
1answer
28 views

Sums involving exponential functions

I am trying to find the closed form of the following related sums: $$(i)\quad\quad S_1(n)= \sum_{m=-\infty}^{m=\infty} |n-m| e^{-p(|n-m|+|m|)} $$ $$ (ii)\quad\quad S_2(n)= ...
2
votes
5answers
141 views

How to prove $\sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0$ [closed]

I would like to prove that: \begin{equation*} \sum\limits_{i=0}^n (-1)^i \binom{n}{i} \binom{n-i}{k}=0;~k\geq0 ; n\geq1. \end{equation*} Can any one help me how to do that? Thanks
0
votes
2answers
79 views

Sigma sign problem from Spivak's calculus text ch 2-2

I need to find a formula for $$ \sum_{i=1}^n (2i-1)^2 = 1^2 + 3^2 + \cdots + (2n-1)^2 $$ This problem is contained in Spivak's calculus ch2-2. I know that: $$ \sum_{i=1}^n i^2 = ...
1
vote
0answers
32 views

Looking for tip/procedure of series solutions to ODE

I have been having a few questions about series solutions to ODE and I found an example that can illustrate my question. It is just a simple example, say we consider the ODE $$ y''-xy'-y=0$$ around ...
3
votes
1answer
69 views

Proving that a trigonometric sum is in $L^2$

How can I use Parseval's identity to prove that $$f(x)=\sum_{k=1}^\infty \frac{\sin(kx)}{1+k}$$ is in $L^2(0,\pi)$? Thank you!
1
vote
0answers
27 views

Estimate from above $\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$

Find an upper bound for $$\sum_{m=1}^{n-1}\frac{1}{n^\alpha-m^\alpha}$$ with $\alpha>1$. I do not know where to start but, for example, if $\alpha=1$ the previous sum is linked to Harmonic numbers ...
0
votes
1answer
29 views

Question about index of summations

I have a question in regard to changing the index of summation. For example, I am confused on why for some problems in my ODE class , such as ' $y''-y'=0$ for example, we suppose the summation of ...
1
vote
1answer
38 views

Find the value of $\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$.

Find the value of $$\sum^{n-1}_{m=1}\left(\frac{1}{n-m}+\frac{1}{n+m}\right)$$ I used WolframAlpha obtaining $$\psi^{(0)}(2n)-\frac{1}{n}+\gamma$$ where $\gamma$ is the Euler-Mascheroni constant and ...
7
votes
0answers
86 views

In how many ways can the integers from $1$ to $n$ be divided into two groups with the same sum?

In how many ways can the integers $1,2,\ldots,n$ be divided into two groups with the same sum? I have tried calculating some of these values for small $n$, but cannot seem to find a pattern. Any ...
1
vote
1answer
33 views

On finite sums and products

I'd like to get a good book on finite summations and products before I study infinite series more in depth next year. The book should cover geometric/ harmonic sums and prove different formulas for ...
3
votes
1answer
74 views

Sophomore's dream changing “x”

"Sophomore's Dream" says $\sum_{n=1}^{\infty}n^{-n}=\int_0^1x^{-x}$ Can you replace the $x$ and $n$ with $2x$ or $x^3$ (and $2n$ or $n^3$) or something? I would guess not, because replacing $x$ with ...
6
votes
2answers
145 views

Proof of an identity of $n!$

I came up (numerically) with an identity concerning n! and I was wondering about a proof of it. Here it is: \begin{align} \ n! &= \sum_{r=0}^{n} { \binom{n}{r} (-1)^r(k-r)^n } \quad \forall n ...
0
votes
1answer
28 views

Differentiating expression involving summation

My problem seemed very simple at glance but I keep missing one term from the answer. Any suggestions? This is the problem: We have $$x_i^* + \xi_i + \frac{\alpha_i}{p_i} \left[ y - \sum_{j=1}^n ...
0
votes
1answer
34 views

Differentiating a sum involving logs

I was doing the problem provided in the picture but I do not understand how do they obtain the answer. I am not sure how to differentiate the sum. I end up getting: alpha - 1 - 1/K. I believe I need ...
1
vote
1answer
38 views

How to express outer sum in a matrix form?

So I have the following equation for a matrix $\mathbf{B}$ given $\mathbf{A}$: $$ b_{ij} = \sum_k \sum_l a_{ki} a_{jl} $$ The question is if there is anyway that I can write that one compactly in ...