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

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4
votes
2answers
345 views

n more each day

It's been a while since I've been at school and I don't work in a field that practices this sort of stuff so I don't know the formula my brain can't wrap my head around the problem. The problem: You ...
0
votes
1answer
39 views

I can't get the answer of this problem about radius of convergence

I have this one: $$ \sum_{n=1}^{\infty} {\frac{\ln{(3n^{2}+5)}x^{n}}{n^2 - 3n +5}} $$ I tried with the classic method: $$ \sum_{n=1}^{\infty} {\frac{\ln{(3n^{2}+5)}((n+1)^2 -3(n+1) + 5)x^{n}}{(n^2 ...
1
vote
3answers
33 views

Compute radius of convergence

Someone can help me with this radius of convergence please? $$ \sum_{n=0}^{\infty}{\frac{3^{2}(5n^{7}+2)}{2^{n}(5n^{3}-1)}}x^{n} $$ I tried $$ r = \lim_{n \to \infty } ...
1
vote
1answer
45 views

Is an integral without a differential component on a finite number of points just a sum?

Is an integral $$\int_{\lbrace 1, 2, 3 \rbrace} f(x)$$ simply the sum $$\sum_{x=1,2,3} f(x)?$$ I ask this question because of the generalization to multiple dimensions of integration by parts ...
0
votes
3answers
42 views

Compute this radius of convergence

Someone can help me with this radius of convergence please? $$ \sum_{n=0}^{\infty}{n^{1/n}x^{n}} $$ I tried $$ r = \lim_{n \to \infty } \frac{n^{\frac{1}{n}}x^{n}}{(n+1)^{\frac{1}{n+1}}(x^{n+1})} ...
0
votes
3answers
43 views

How do I finish this summations problem?

I have posted a picture since I don't know how to make the summation symbols with the lower and upper summations on keyboard, sorry about that.. $$\sum_{a=1}^9\sum_{b=0}^9(101a+10b)$$ The answer is ...
-1
votes
1answer
66 views

Rewrite and approximate the sum as an Integral $\sum_{i=1}^{1000} \sqrt{i}$ [closed]

This is not an Infinite sum !, how do we change this to an Integral. $ $ We normally write an integral as an infinite sum.
0
votes
1answer
35 views

Comfirmation of third derivative of symbolic equation including summation

With previous help I was able to find the first derivative of an equation for a work project. Now I'm after the second and third derivative, for use in a program to find the maximum (Which I must do ...
0
votes
0answers
44 views

Intuition behind summation [duplicate]

$$\sum_{n=1}^{\infty}n = -1/12$$ Could someone give me an intuitive explanation of why this is true? I just completed Calc 2 and finished a unit on series, and according to the stuff we learn't the ...
5
votes
1answer
62 views

How many decimal representations are possible for the number 1

I know that there at least two $0.\overline{9}$ and 1 Is there a neat and more concrete way to understand this problem.
4
votes
2answers
74 views

How to evaluate the sum $\sum_{k = 0}^{n}2^k {{n}\choose {k}}$ [duplicate]

How do I evaluate the sum: $$\sum_{k = 0}^{n}2^k {{n}\choose {k}}$$ I know that $2^k = {n \choose 0} + {n \choose 1} + {n \choose 2} + {n \choose 3}... {n \choose k}$, but I don't know how to proceed ...
11
votes
1answer
80 views

Convergence of $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ where $f(n)$ is the number of prime divisors

Let $f(n)$ be the number of prime divisors of a number $n$ counted with their multiplicities. Show that the series $\sum_{n=1}^{+\infty}\frac{(-1)^{f(n)}}{n}$ converges and has sum $0$. Attempt ...
1
vote
1answer
116 views

Find the sum of the series $\sum \limits_{n=3}^{\infty} \dfrac{1}{n^5-5n^3+4n}$

Feel free to skip obvious steps, or use a calculator when required. I just want to understand the theme of the solution. Any help is appreciated EDIT : We can write$$ \dfrac{1}{n^5-5n^3+4n} = ...
0
votes
2answers
31 views

If a sequence $\{a_n\}$ satisfies the Inequality $a_{n+1} < ka_{n}$, then show that $ \lim\limits_{n \to \infty} a_n =0$ where $0< k , a_n< 1$

I know one solution. Consider $\sum a_n$ Then use ratio test to show that the series converges, hence the sequence. Any other Ideass !
1
vote
3answers
63 views

Find all values of $c$ for which the following series converges $\sum_{n=1}^{\infty} \left(\dfrac{c}{n}-\dfrac{1}{n+1}\right)$

I know that the series in question converges when $c=1$, but I have no concrete way to find all such values of $c$ for which this is true.
1
vote
1answer
16 views

Proving an identity involving binomial coefficients and fractions

I've been trying to prove the following formula (for $n > 1$ natural, $a, b$ non-zero reals), but I don't know where to start. $$\sum_{j=1}^n \binom{n-1}{j-1} \left( \frac{a-j+1}{b-n+1} \right) ...
3
votes
1answer
52 views

Give example of a series $\sum a_n$ such that the series is conditionally convergent. and $\sum na_n$ is convergent

I tried all the conditionally convergent series I know, I found $\sum na_n$ to be diverging for all of them. But I am sure the question is correct
7
votes
1answer
203 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
60 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
40 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
19 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
112 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
42 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 ...
2
votes
0answers
98 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 ...
0
votes
1answer
237 views

Solution of Introduction to Algorithms 3rd edition (Cormen)

I'm studying algorithms by Cormen book. I have already find a pdf with the answers of the questions. www2.compute.dtu.dk/~phbi/files/teaching/solution.pdf But it isn't have the all solutions. I'm ...
0
votes
5answers
61 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
40 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
29 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
37 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
54 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
32 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
45 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
68 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
66 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
29 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
48 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
19 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
23 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
29 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
23 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
40 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} + ...
9
votes
3answers
282 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
164 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
25 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
19 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
30 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
48 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
47 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
217 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
43 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 ...