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

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1answer
19 views

Laplace pairs - proof of summation transform

I am studying this question for my finals revision and I'm lost on how to start it, can anyone suggest something? Probably pretty simple but I've hit a dead end. Here's the question: If $F_i(t)$ ...
2
votes
4answers
106 views

Direct Proof for sum of $n$ integers equation?

I am trying to prove by direct proof that $$3+5+7+\ldots+(2n+1)=n(n+2)$$ for all natural numbers $n$. I figured out how to do it by induction, but I know it can be done directly and I can't ...
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1answer
31 views

Can someone draw a plot for this function?

Can someone draw a plot for this function? $ f(x) = \begin{cases} \sum_{i=2}^{x}\left(\frac{\prod_{k=1}^{i-1}\left(2k-1\right)\,\cdot\,-\left(-\frac{1}{2}\right)^{i}}{i!}\right) + \frac{3}{2} & x ...
1
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1answer
56 views

$a_n = b_n -b_{n-1}$ Prove that $\sum_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists

Let $\{b_n\}$ be a sequence Let $a_n = b_n - b_{n-1}$. Prove that $\sum\limits_{n=1}^{\infty} a_n$ converges iff $\lim_{n \to \infty} b_n$ exists. I am extremely stuck on this homework problem and ...
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0answers
15 views

expression for a rational number

Let $x=\sum_{k=1}^n-b_kg^{-k}$ where $g\in \Bbb N$ and $b_k\in\{0,\cdots,g-1\}$. I want to write $x$ in the form $x=m+\sum_{k=1}^nc_kg^{-k}$ with $c_k\in\{0,\cdots, g-1\}$ and $m\in \Bbb Z$. Doing ...
1
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0answers
38 views

Can you get the average order of $ \left( 1+|\mu(n)| \right)^{M(n)} $, where $\mu(n)$ and $M(n)$ are the Möbius and Mertens functions, respectively

When yesterday I was interested in do a little study about the arithmetic function $$f(n)=\left( 1+|\mu(n)| \right)^{M(n)},$$ defined for integers $n\geq 1$, which $\mu(n)$ is the Möbius function and ...
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2answers
33 views

Find general formula for $\sum _{i=1}^{n} \frac {(-1)^i i}{(2i-1)(2i+1)}$

I was able to find formulas for simpler expressions but I can't find the general formula for this one: $\sum _{i=1}^{n} \frac {(-1)^i i}{(2i-1)(2i+1)}$ I don't see any particular trend that would ...
0
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2answers
23 views

General formula for a summation

I can't find the general formula for the following sum. $q \in \Bbb R, q \ne 1$ $\sum _{i=0}^{n} q^{2i}$ Any hints?
1
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1answer
34 views

How to expand the summation term with power?

How to expand the following: $$ \left( \sum^{M}_{m=0} \frac{x^{m}}{m!} \right)^{K} $$ where $M$ and $K$ are positive integers.
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1answer
73 views

$\sum_{n=1}^{\infty} ne^{-2n}$ estimate to 4 decimal places

I am supposed to estimate the sum correct to 4 decimal places and assume it converges. I know that I am supposed to plug in numbers for $n$ (Instructor says that solving for $n$ is impossible) however ...
1
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1answer
15 views

Solution check: summation inequality proof by induction

I'm not sure if what I've done works or if it's proof enough. (I need to prove that the inequality is true $\forall n \in \mathbb{N}$). $\sum_{i=n}^{2n} \frac{i}{2^i} \leq n$ $P(1)$ works. I assume ...
0
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1answer
35 views

Summation of $\int\frac{1}{1+x}dx$ in a range of 1 to infinity. [closed]

Let $0 < \alpha < \beta < 1$. Then $$\sum_{k=1}^\infty \int_{1/(k+\beta)}^{1/(k+\alpha)} \frac{1}{1+x}dx$$ is equal to $$ \begin{align} &(A)\ln \frac{\beta}{\alpha}\qquad\qquad ...
0
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1answer
46 views

If $\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $ then…

If $$\ (1+3x+x^2)^{10}=\sum_{r=0}^{20}a_r x^r\ $$ Then then what is the least number except 1 which divides the following:$$\ \sum_{r=0}^{20}(3r+1)a_r\ $$ EDIT: i have put x=1 then it is something ...
2
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3answers
65 views

$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $

Let $\omega \ $be a root of the polynomial $\ x^{2016} +x^{2015}+x^{2014}+...+x+1=0 \ $. Then find the value of the following sum: $$\sum_{k=0}^{2016}(1+ \omega^k)^{2017}\ $$ Well I have simplified ...
2
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2answers
45 views

Summation of an expression $\sum_{h=0}^{\ln n}\frac{h}{2^h}$

How can be we get the closed form for this expression? $$ \sum_{h=0}^{\ln n}\frac{h}{2^h} $$
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0answers
24 views

On the zeroes of sine complex function, and a search for a special sequence, following Riemann's approach

If there are no mistakes from the Fourier expansion series for the fractional part function we can write, using a substituion, that for $1<x<e^2$ with uniform convergence $$\frac{1}{2}\log ...
0
votes
1answer
19 views

Can we equate general term of two equal summations provided limit goes from $0$ to $\infty$?

For example: If $∑_{x=0}^{\infty}f(x)=∑_{x=0}^{\infty}g(x)∑_{x=0}^{\infty}f(x)=∑_{x=0}^{\infty}g(x)$, can we say $f(x)=g(x)f(x)=g(x)$ for all $x$, or is there a possibility that they are not ...
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1answer
21 views

Understanding a text in a book about the estimation

Now $$e_n-e_0=\sum_{k=0}^{n-1}\left [ -\frac{1}{12(k+x)^2}+\mathcal{O}\left ( \frac{1}{(k+x)^3} \right ) \right ]; \tag{*}$$ therefore, $\lim_{n\to\infty}e_n-e_0=K_1(x)$ exists. Set ...
1
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1answer
50 views

How to calculate a sum using a geometric series

How to calculate this with a simple calculator. sum_{i=20}^n=59 0.1*600*1.04^(60-i) = ? I tried this but it's wrong. Can somebody please tell me where I made a ...
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1answer
18 views

$\sum_{1 \leq i,j \leq n, i|n, j|n} gcd(i,j)$

$$S = \sum_{1 \leq i,j \leq n, i|n, j|n} gcd(i,j)$$ I can't find a way to solve this. Can I find a general formula or a way to solve this?
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0answers
31 views

I am looking a comparison of this computation and Riemann's approach for $lcm(1,2\ldots,x)$

Looking a comparison with a reasoning due to Riemann, I ask to me about the behaviour as $x\to\infty$ of the following arithmetical function $$ \left( \prod_{n\leq x}n^{-\mu(n)}\right)\cdot \left( ...
2
votes
2answers
48 views

Three nested summations

I'm not sure of how to solve three nested summations and I came up with the following. Is it wrong? $$\sum\limits_{i=1}^n\sum\limits_{j=1}^n\sum\limits_{k=1}^{2i+j} ...
7
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1answer
104 views

how to calculate double sum of GCD(i,j)?

I stumbled upon a programming question which wanted me to calculate : $$G(n) = \sum _{i=1}^{n} \sum _{j=i+1}^{n} gcd(i, j).$$ now I wrote a code to solve this problem but it takes polynomial time to ...
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2answers
37 views

Big O for a $\cos$ series

I have to show that $ \sum_1^N \cos(nx) = O(\frac 1{|x|}), [-\pi, \pi] $, x different from 0. I really don't know how to show that. I obviously know that $\cos(nx)$ is bounded by $1$, I know what ...
2
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0answers
31 views

Limiting distribution of infinite sum of weighted bernoulli?

Let $p_n$ be some fixed pulse, for example $p_n =e^{-n^{2}}$ We have an infinite sum $y = \sum_{n=-\infty}^{\infty} a_n p_{-n}$ where $a_n$ are iid bernoulli random variables taking the values $+/- ...
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2answers
58 views

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$?

How to evaluate $\lim\limits_{x\to 0}\frac{e^{\arctan{(x)}}-xe^{\pi x}-1}{(\ln{(1+x)})^2}$? So I think we expand to $x^2$ since the lowest term for $\ln(1+x)$ is $x$ Let $u=\arctan{(x)}$ ...
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2answers
22 views

How to compute $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}$?

How to compute $\sum^{\infty}_{n=2}\frac{(4n^2+8n+3)2^n}{n!}$? I am trying to connect the series to $e^x$ My try: ...
0
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1answer
18 views

Summation combined with limits

$$\lim_{n\to\infty}\sum_{r=1}^{n} ((r+1)\sin(π/r+1)-r\sin(π/r)$$ What I have tried so far I have posted in the image can you please tell me what to do from there on http://i.stack.imgur.com/6exCz.jpg ...
1
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0answers
32 views

Products of $k^{\mu(k)}$, where $\mu(n)$ is Möbius function, and the Prime Number Theorem

We can write $$e^{-\Lambda(n)}=\prod_{d\mid n}d^{\mu(d)},$$ where $\mu(n)$ is the Möbius function and thus $\Lambda(n)$ is von Mangoldt's function. Then taking the product from $1$ to $N$ we've for ...
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2answers
38 views

Summation over a floor function of a first degree polynomial

I've been trying to solve a difficult programming question for the last four days. I've gotten most of it done, but the piece I can't seem to figure out is this: Find a closed form expression of ...
2
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2answers
36 views

Induction proof of the identity $\cos x+\cos(2x)+\cdots+\cos (nx) = \frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}$ [duplicate]

Prove that:$$\cos x+\cos(2x)+\cdots+\cos (nx)=\frac{\sin(\frac{nx}{2})\cos\frac{(n+1)x}{2}}{\sin(\frac{x}{2})}.\ (1)$$ My ...
0
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1answer
19 views

Summation of a fraction containing a summation operator

I came across a proof which had the following sequence: $$\sum_{i=1}^n k_i y_i = \sum_{i=1}^n \frac{(x_i - \bar x)y_i}{\sum_{j=1}^n (x_j - \bar x)^2}$$ where $$k_i = \frac{(x_i - \bar ...
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0answers
16 views

Is the following correct way of manipulating taylors series?

For $\sum^{\infty}_{n=1}\frac{(-1)^{n}\pi^{2n}}{4^n(2n+1)!}$. Let $x=\frac{\pi}{2}$, the series becomes ...
2
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2answers
28 views

How can I prove the concavity of $f(p_1,p_2,\ldots,p_n) = \sum_{i = 1}^n p_i(1-p_i)$?

Assume $p_n$ is the probability of being in class $n$ which mean that $f(0) = 0$ , $f(1) =0$ , and $p_1+p_2 = 1$ I need to come up with a concave function that show the relation between $p_1$ and ...
2
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2answers
38 views

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$?

How to compute taylor series $f(x)=\frac{1}{1-x}$ about $a=3$? It should be associated with the geometric series. Setting $t=x-3,\ x=t+3$, then I don't know how to continue, could someone clarify the ...
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1answer
34 views

How to compute $\sin{(\pi x)}$ about $\frac12$ in taylor series?

The correct answer is supposed to be $\sum\frac{(-1)^n}{(2n)!}\pi^{2n}(x-\frac12)^n$ which I don't understand. Since the function is about $x=\frac12$, so $(x-\frac12)^n$ is good. But ...
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1answer
26 views

Finding The maximum and minimum value of a summation given a condition

If $a_i = 1 - \frac{1}{N_i}$ and $\sum\limits_{I=0}^k{N_i} = n$, then what is the maximum and minimum values of $\sum\limits_{I=0}^k{a_i}$? Please help, I've tried to solve it but then I got confused. ...
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2answers
94 views

How to calculate the series $-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}…$?

$-\frac12-\frac14+\frac13-\frac16-\frac18+\frac15-\frac{1}{10}...$ After rearrangement the series looks like $\sum^{\infty}_{n=2}\frac{(-1)^{n+1}}{n}$. My way of doing this is using Taylor series of ...
0
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1answer
18 views

Given the estimator find wheter

X has an uniform distribution on interval $(0,\theta]$ where $\theta$ is a positive parameter Given the estimator: $$T(X_1,X2, \ldots, X_n)=\frac{2}{n} \sum_{i=1}^n X_i$$ Find whether this estimator ...
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2answers
44 views

Prove that $(a^n - b^n) = (a-b) \sum_{i=1}^n a^{i-1} b^{n-i}$

Let it be $a, b \in\Bbb R$. Prove that $\forall n \in\Bbb N$, $(a^n - b^n) = (a-b) \sum_{i=1}^n a^{i-1} b^{n-i}$. Deduce the formula of the geometric sum: $\forall a ≠ 1, \sum_{i=0}^n a^i = ...
0
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1answer
54 views

Maximizing $x_1 x_2 x_3\cdots x_m\left(\frac{1}{x_1}+\frac{1}{x_2}+\frac{1}{x_3}+\ldots + \frac{1}{x_m} - 1\right)$

$x_1, x_2, x_3, ... ,x_m > 0 \quad \forall m\geq 3$ , $x_1+x_2+x_3+...+x_m = 1$. What is maximum of $x_1 \cdot x_2\cdot x_3\cdot ... \cdot ...
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0answers
35 views

Use the Fourier Series of $f(x)=x^2+1$ to find the sum of the series

I have found the Fourier Series of $f\left(x\right)=x^{2}+1$ on the interval $\left[-\pi, \pi\right]$ extended periodically to $\mathbb{R}$ to be $$ ...
7
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6answers
553 views

Consecutive integers sum with different steps

First of all: beginner here, sorry if this is trivial. We know that $ 1+2+3+4+\ldots+n = \dfrac{n\times(n+1)}2 $ . My question is: what if instead of moving by 1, we moved by an arbitrary number, ...
1
vote
3answers
63 views

Proof $\sum \frac{1}{n^a}$ is convergent for a > 1

I get to the fact that $\sum_{k=1}^n \frac{1}{k^a}$ < $\frac{1}{a-1} - \frac{1}{(a-1)(n+1)^{a-1}} - \frac{1}{(n+1)^a} + 1$ and hence $\sum_{k=1}^n \frac{1}{k^a}$ is bounded. How to deduce $\sum ...
0
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2answers
14 views

Proof $\sum_{k=1}^n \frac{1}{(k+1)^a}$ < $\frac{1}{a-1} - \frac{1}{(a-1)(n+1)^{a-1}}$

I also know that $\frac{1}{a-1} - \frac{1}{(a-1)(n+1)^{a-1}}$ = $\sum_{k=1}^n \frac{1}{(a-1)k^{a-1}} - \frac{1}{(a-1)(k+1)^{a-1}} $ Any hint or help to solve this please?
0
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0answers
16 views

How to determine a formula for an index of compliancy

I hope you will be patient with the inarticulate question of a non-mathematician. It's hard to get an answer when you don't even know how to ask the question. Here the contest: Let's say that I have ...
0
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3answers
39 views

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series?

How to compute $\lim\limits_{x\to 0}\frac{e^{x^2}-\cos{(2x)}-3x^2}{x^2\sin{x^2}}$ limits by using taylor series? I think that we need to take every familiar taylor series (i.e. $e^x,\sin{x}$) and ...
3
votes
2answers
54 views

Finding Exact Values of Specific Infinite Series

Prove that $\Sigma_{n=1}^{\infty}(n/2^n)=2$ and that $\Sigma_{n=1}^{\infty}(n^2/2^n)=6$. Thoughts: I have a feeling that if someone shows me how to do one, I'll be able to figure out the other. So ...
0
votes
0answers
22 views

How to test the convergence of the following sereis?

$\sum^{\infty}_{n=1}\sin{\frac{1}{n}}$: The only test I can think of for this one is basic comparison ($\sin{\frac{1}{n}}\le\frac{1}{n}$). But $\frac{1}{n}$ diverges. ...
1
vote
1answer
53 views

Summing $n$ numbers so that they equal $0 \mod{n}$

Let $A_n=\{(a_1,a_2,\dots,a_n) :\sum_{i=1}^na_i=0\mod{n}\}$, where $a_i\in[n-1]$. How many elements are in $A_n$? My initial attempt was a stars-and-bars argument. For example, let $n=4$. Then we ...