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

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7
votes
1answer
210 views

Series $\sum \frac{1}{n^2\sin^3n}$

Question : Show that series $\sum \cfrac{1}{n^{2}\sin^{3}n}$ is divergent. Hint: Show that $$\sum \frac{1}{n|\sin(n)|}$$ is divergent. I am interested in other possible proofs for this question. ...
0
votes
1answer
75 views

Concerning the sum $\sum_{n = 1}^\infty \sin nx$

I recently came across this question and I posted an answer. It has been pointed out that my answer is incorrect. I cannot work out what is wrong with my reasoning. The answer I gave corresponds with ...
1
vote
1answer
42 views

How do i evaluate this sum?

Let $[x]$ be the nearest integer to $x$. (for $x=n+\frac{1}{2}, n \in N$, let $[x]=n$). Find the value of $$\displaystyle\sum_{m=1}^{\infty} [\sqrt m]^{-3}$$
0
votes
1answer
26 views

Limits on repeated sum in circle method

In Bob Vaughan's book The Hardy-Littlewood Method, early on he gives a sum \begin{equation} \left(\sum_{m=1} ^N e(\alpha m^k)\right)^s = \sum_{m_1 = 1} ^N \sum_{m_2 = 1} ^N \cdots \sum_{m_s = 1} ^N ...
3
votes
6answers
127 views

How can I show that $\sum_{k=0}^{\infty} \frac{k-1}{2^k} = 0$?

I'm studying the algorithms book and I have a doubt. I don't know how can I prove this summation: $$\sum_{k=0}^{\infty} \frac{(k-1)}{2^k} = 0$$
2
votes
1answer
67 views

Stirling-like sum equal to zero when $k>n$

I need to prove that $$\sum_{r=0}^k\binom{k}{r}(-1)^r r^n=0$$ when $n<k$. I know that the formula above can be easily transformed into the Stirling number of the Second kind formula, which is ...
3
votes
0answers
176 views

Find the limit of $\sum \frac{1}{log^n(n)}$

Working on convergence and divergence of infinite series, I recently focused my attention on the summation $$\displaystyle\sum\limits_{n=2}^{\infty} \frac{1}{log^n(n)}$$ While proving the convergence ...
1
vote
2answers
11k views

Show that $\sum_{k=0}^{\infty}k^2x^k= \frac{x(1+x)}{(1-x)^3}$ for $0 < |x| < 1$

Show that $$\sum_{k=0}^{\infty}k^2x^k= \frac{x(1+x)}{(1-x)^3}$$ for $0 < |x| < 1$. (This is appendix question A.1-3 from Introduction to Algorithms by Cormen)
0
votes
5answers
67 views

How do I find the partial sum of this

So I have a sum defined below: $$ \sum_{m=1}^n 2^{-m} $$ I know the partial sum equals $$ \frac{1}{2^n}(2^n - 1)\ $$ But how do you go from one to the other?
1
vote
1answer
78 views

Algebraic manipulation of floors and ceilings

I am trying to solve the summation $$ \sum_{n=3}^{\infty} \frac{1}{2^n} \sum_{i=3}^{n} \left\lceil\frac{i-2}{2}\right\rceil $$ I will list some of the simplifications that I've found so far, and ...
2
votes
1answer
33 views

Does $\sum_{i=1}^{k-1}\lceil \log_2\frac{N}{i}\rceil$ have a closed form?

Does the following have a closed formula? $$\sum_{i=1}^{k-1}\left\lceil \log_2\frac{N}{i}\right\rceil$$
0
votes
1answer
41 views

What is the value of $\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n[\sqrt\frac{4i}{n}]$

What is the value of $$\displaystyle{\lim_{n\to \infty}\frac{1}{n}\sum_{i=1}^n[\sqrt\frac{4i}{n}]}$$ Where [x] denotes the greatest integer less than or equal to x Answer is given as $3$ but I think ...
0
votes
4answers
59 views

Evaluate $\displaystyle{\lim_{n\to \infty}\frac{1}{n}\sum_{r=0}^{n-1}\cos\frac{r\pi}{2n}}$

What is the value of $$\displaystyle{\lim_{n\to \infty}\frac{1}{n}\sum_{r=0}^{n-1}\cos\frac{r\pi}{2n}}$$ The answer is given as $1$ but I am almost 100% sure it will not be $1$. What will be the ...
2
votes
3answers
36 views

How to solve $\frac{1}{n}\left[1+2\sum_{k=1}^{n-1}\frac{1}{\sqrt{\frac{n}{n-k}}}\right]$

I want to find an analytical expression for: $\frac{1}{n}\left[1+2\sum_{k=1}^{n-1}\frac{1}{\sqrt{\frac{n}{n-k}}}\right]$ I know that the result is independent of $n$ when $n$ is large, because I ...
0
votes
2answers
58 views

Evaluate $S=\Sigma\Sigma\Sigma x_ix_jx_k$

I am a novice in this type of sums and I can't even understand the meaning of the three sigmas. Somehow, I am guessing that the answer might be $0$ but I am not sure. I need a well-explained ...
5
votes
1answer
74 views

Calculate the sum of S.

Consider $n\in\mathbb{N}.$ Find the sum of:$$S=\left(\dfrac{C_n^0}{1} \right)^2+\left(\dfrac{C_n^1}{2} \right)^2+\cdots+\left( \dfrac{C_n^n}{n+1}\right)^2$$ I don't know how to solve it, i don't ...
3
votes
2answers
79 views

Integer sum as binomial coefficient

What's the rule for expressing integer sums as binomial coefficients? That is, for $p=1$ it is $$\sum_{n=1}^N n^p = {{N+1}\choose 2} $$ What is it for higher powers?
1
vote
1answer
58 views

Proving Product of Transition Matrices is again a Transition Matrix.

Let $P = [p_{ij}]$ be an $n\times n$ transition matrix for an $n$-state markov chain. How do you prove that $P^2$, or even better, that $P^n$ is again a transition matrix? My approach leaves me ...
1
vote
1answer
57 views

A problem of sum floors

let $n$ be a positive integer, prove that $$\sum_{i=0}^{\left\lfloor\frac{n}{3}\right\rfloor}\left\lfloor\frac{n-3i}{2}\right\rfloor=\left\lfloor\frac{n^2+2n+4}{12}\right\rfloor.$$ It looks like we ...
1
vote
0answers
20 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
1
vote
1answer
28 views

$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$

I have found this equality: $$\prod_{i\in I}(1+x_i)=\sum_{J\subseteq I}\prod_{j\in J}x_j$$ Do you think is it true?
1
vote
2answers
32 views

Proving that mean KDR in a videogame is one

This is not related to schoolwork. A friend of mine challenged me to prove that the mean KDR (assuming players can only die at the hands of other players) must always be equal to one. I have gotten ...
1
vote
1answer
33 views

A sum of Laguerre polynomials

I'm looking to find a closed-form expression for the sum $$S = \sum_{n=0}^N e^{-x/2} L_n^{0}(x),$$ where $L_n^{0}$ is the $n$th Laguerre polynomial. Using the formula $$L_n^{\alpha}(x) = \sum_{m=0}^n ...
0
votes
2answers
66 views

How to derive these inequalities?

I can derive the inequalities $$ n^p < \frac{(n+1)^{p+1} - n^{p+1}}{p+1} < (n+1)^p $$ for any positive integers $p$ and $n$. These actually follow from the identity $$b^p - a^p = (b-a)(b^{p-1} + ...
14
votes
3answers
344 views

Proving that $ \displaystyle \gamma = \int_{0}^{1} \!\!\int_{0}^{1} \!\frac{x - 1}{(1 - x y) \log(x y)} \, \mathrm{d}{x} \, \mathrm{d}{y} $.

In 2005, J. Sondow found a surprising formula for the Euler-Mascheroni constant $ \gamma $. The formula is $$ \gamma = \int_{0}^{1} \int_{0}^{1} \frac{x - 1}{(1 - x y) \log(x y)} ~ \mathrm{d}{x} ~ ...
4
votes
3answers
173 views

Infinite Sum of algebraic expression

Prove that $$\sum_{i=1}^{\infty} \frac{1}{i(2i+3)} = \frac89 -\frac23\ln2$$ I tried using integration but failed miserably. Hints please.
0
votes
2answers
28 views

Sums Convergence tests

$ \sum_{k=1}^\infty k(\frac 14)^k $ i've tried to do the D'Alembert's criterion and i got $ \frac 14 $ but according to wolfram alpha the answer is 4\9 thanks
2
votes
2answers
23 views

How to obtain this upper bound on the summation from this inequality?

I can show that $$ \frac{1}{\sqrt{n}} < 2 (\sqrt{n} - \sqrt{n-1} ) $$ for $n \geq 1$. Now from this how to derive the following inequality? $$ \sum_{n=1}^m \frac{1}{\sqrt{n}} < 2\sqrt{m} - 1 $$ ...
0
votes
2answers
57 views

Samplification of a sum of multiplication

Supposing I have the following sequence based on two indexes: $a$ and $b$. For $a$ starting with $1$ and $b$ starting with $5$ we have the following sum: $$1 \cdot 5 + 2 \cdot 4 + 3 \cdot 3 + 4 ...
1
vote
1answer
110 views

An algorithm to find X numbers that sum up to a given value

I have this little problem and I was wondering if some mathematician here knew something useful about how to solve this or even how to approach this right. In the simplest terms I have a set of ...
1
vote
1answer
52 views

Finding the limit of an integral

Evaluate $$\displaystyle\lim_{j\rightarrow \infty} \displaystyle\int_{0}^{a} \frac{1}{j!} \left(\ln \left(\frac{A}{x}\right)\right)^{j}dx$$
4
votes
1answer
256 views

Is there a closed form expression for the sum of all the proper divisors of an integer?

I have already found a summation formula here: http://math.stackexchange.com/a/22723, and also a very interesting recursive formula here: http://math.stackexchange.com/a/22744. Any ideas on how to ...
0
votes
1answer
55 views

Upper bound for the sum $ \sum_{k=1}^N \frac{1}{\varphi(k)}$

Is there an upper bound for the sum $$ \sum_{k=1}^N \frac{1}{\varphi^{\alpha}(k)} $$ where $\varphi(n)$ is the Euler totient function and $\alpha\geq 1$ a real constant? In particular, I'm interested ...
1
vote
0answers
102 views

What is the sum of Psi/Digamma-function of consecutive arguments? Is there a closed form?

In a consideration of summation of a series $$ s = a_0 + a_1 + a_2 + \cdots \tag 1$$ with $$\lim_{k \to \infty} a_k=0$$ but slowly decreasing, the coefficients $a_k$ are somehow related to $1/k^2$ ...
0
votes
1answer
50 views

Double sigma summation is in complexity calculation

Basically i was reading skiena and doing exercise of 2nd chapter.The result of my calculation got differed from the actual solution given on Solution site and there is one thing i don't understand how ...
3
votes
4answers
146 views

Proving Combinatorical Summation: $n!=\sum_{k=0}^n(-1)^k\binom{n}{k}(n-k)^n$ [duplicate]

been stuck with this question for the last few hours, any help would be appreciated. $$ {\large n! = \sum_{k = 0}^{n}\left(-1\right)^{k}{\,n\, \choose \,k\,} \left(\,n - k\,\right)^{n}} $$ what I ...
1
vote
0answers
24 views

Solving sum of one variable with real exponents

I'm working with an annoying maximisation problem at the moment. I've spent a long time Googling, but I'm not having much success and I suspect it would be simple enough if I had the right tools. I ...
0
votes
1answer
51 views

Proving $\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$

Prove that $$e^{i\theta}\cdot\frac{e^{in\theta}-1} {e^{i\theta}-1}=\frac{1}{2}\left(e^{in\theta}-e^{-in\theta}\right) +\frac{1}{2}\left(e^{in\theta}+e^{-in\theta}\right)\\$$ I tried to use $$ ...
0
votes
2answers
58 views

Proving that $\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$.

Prove: $$\sum^n_{k=1} e^{ik\theta}=\sum^n_{i=1}\cos k\theta +i\sum^n_{k=1}\sin k\theta$$ Thanks a lot!! I tried: With Euler's identity I can get $\sin x= \dfrac{e^{ix} - e^{-ix}}{2i}$ and the ...
0
votes
1answer
25 views

Average sum of seria

I need some help with the next question in probability: In the range {1,2,...,100}, someone picks randomly 15 different numbers, with the same probability for each number. What is the average sum ...
28
votes
1answer
601 views

Evaluating the sum $\lim_{n\to \infty}\sqrt[2]{2+\sqrt[3]{2+\sqrt[4]{2+\cdots+\sqrt[n]{2}}}}$

The following nested radical $$\lim_{n\to \infty}\sqrt{2+\sqrt{2+\sqrt{2+\cdots}}}$$ is known to converge to 2. We can consider a similar nested radical where the degree of the radicals increases: ...
1
vote
1answer
47 views

$\sum_{x=a}^{b-1}\frac{1}{x}$ and $\sum_{x=a+1}^b\frac{1}{x}$

I have to prove the following relations: $\sum_{x=a}^{b-1}\frac{1}{x}\geq\log b - \log a $ $\sum_{x=a+1}^{b}\frac{1}{x}\leq\log b - \log a $ I tried to use the relation that $\int_a^b \frac{1}{x} ...
0
votes
2answers
25 views

Convergence of sum using D'Alembert.

I have to find the convergence of this series: $$\sum \limits_{n=0}^{\infty} \frac{(1+{\frac 1n})^n}{2^n}$$ I started by using D'Alembert: $\lim \limits_{n \to \infty} \frac{a_{n+1}}{a_n}$, So : ...
2
votes
0answers
35 views

statements about summation

Could you help me prove this statements about summation? I know that the second prove is easy of be written, but can I put that summation before cos(theta) and sin(theta)? yes. But why? Do you ...
0
votes
1answer
28 views

Multiplying infinite sums

Is that true for infinite sums that $c\cdot \sum_{n=1}^{\infty} a_n=\sum_{n=1}^{\infty} c\cdot a_n$?Or it only applies when the sum is finite($\sum a_n \lt \infty$)?
0
votes
0answers
94 views

Derivative of Log of Summation of exponential function (base e)

A financial formula that I am implementing requires that I find the first derivative of a function to find a local maxima, from scratch. Can someone please help me with finding the first derivative of ...
3
votes
3answers
107 views

Why is $\frac{\sum_{n=1}^{\infty} n}{\sum_{n=1}^{\infty} n}$ indeterminate?

We all know that $\dfrac{f(x)}{f(x)} = 1$ (where $f(x) \neq 0$) and that $\sum_{n=1}^{x} n = \dfrac{x(x+1)}{2}$. So, given $f(x) \stackrel{\text{def}}{=} \sum_{n=1}^{x} n$, we show that ...
1
vote
1answer
36 views

Discrete math: Sum of Geometric series on a problem - Notes make little sense.

I've been reading a PDF of slides from my Discrete Math I professor. The title is Sums, Products and Asymptotic Estimations. He gives us a problem to fire off the lecture, which is the following: ...
1
vote
1answer
39 views

Prove summation related to cycles

Let $b_r(n,k)$ be the number of n-permutations with $k$ cycles, in which numbers $1,2,\dots,r$ are in one cycle. Prove that for $n \geq r $ there is: $$ \sum_{k=1}^{n} ...
1
vote
2answers
56 views

Summation closed form.

I have tried to figure out how to get the close form of: $$\sum_{k=0}^n k^22^{n-k}.$$ I tried to write down each number of the summation but couldn't find any thing to do with that.. can please ...