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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2answers
27 views

What is the sum $\sum_{k=0}^{n-1} e^{kx}$?

My Precalc teacher gave me this as a question and I simply cannot figure out how to do it.
0
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0answers
7 views

Multiple Variables in an Infinite Sigma notation equation.

If I have an equation using sigma notation that contains two variables, say $n$ and $j$. $j = 0$ to start, and while $j < \infty $ and $n \le 1$; what is the sum. I was curious how much you would ...
1
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0answers
6 views

The autocovariance function of ARMA(1,1)

So I am reading Brockwell and Davis introduction to Time Series analysis on page 89 where he derives the ACVF of an $ARMA(1,1)$ given by: $X_t - \phi X_{t-1}=Z_t+\theta Z_{t-1}$ with ${Z_t}$ is ...
0
votes
1answer
27 views

Limit-Sum problem.

Here's a limit/sum problem I dreamt up: $$\lim_{a\to \infty}\sum_{n=1}^{\infty}\frac na$$ I have a feeling there is a simple solution, but I'm not sure. Apologies if this question doesn't include ...
1
vote
1answer
16 views

How many $(r+1)$- subsets of $[n+1]$ have $(k+1)$ as their largest element?

Let $[n+1]$ be the set defined by $[n+1]=\{1,2,\ldots,n+1\}$. Call a subset of $[n+1]$ with $r+1$ distinct elements an $(r+1)$-subset. How many $(r+1)$-subsets of $[n+1]$ have $(k+1)$ as their ...
1
vote
1answer
33 views

How do I display this as a sum?

How do I write this infinite series using the sigma notation? $$1+f'(n)m+\frac{f''(n)}{2!}m^2f(n)+\frac{f'''(n)}{3!}m^3(f(n))^2+...$$ My attempt: ...
3
votes
1answer
59 views

An identity involving Bernoulli and Stirling numbers

I was playing with some combinatorial sums and made an observation that I didn't know how to prove: $$\forall n\in\mathbb N,\hspace{10px}\sum_{k=1}^n\frac{B_k\ ...
0
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0answers
26 views

Convert Infinite integral to sum

I want to convert an infinite integral to sum. I could not find much info on this online as my integral is from $\infty$ to $-\infty$. For example how would you convert the following? ...
1
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1answer
38 views

Prove $\sum \frac{t}{(1+y)^t }= \frac{y+1}{y^2}$

I see on Wolfram Alpha that $\sum \frac{t}{(1+y)^t} = \frac{y+1}{y^2}$ when t goes to infinity. I cannot, however, proove it myself. What theory is used and how do I start the proof?
0
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1answer
20 views

How do I expand/solve the following summation? [duplicate]

$\sum\limits_{i=1}^{n-1} i$. I know the answer is $\frac{1}{2}(n-1)n$ but I don't quite understand it how to get there.
5
votes
0answers
41 views

Summation of a function

Let $n$ is a positive integer. $n = p_1^{e_1}p_2^{e_2}...p_k^{e_k}$ is the complete prime factorization of $n$. Let me define a function $f(n)$ $f(n) = p_1^{c_1}p_2^{c_2}...p_k^{c_k}$ where $c_k = ...
2
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4answers
115 views

Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $

The following series converges to 3/2 but I do not know why. $$ \sum_{n=1}^\infty \frac{n^2}{3^n} = \frac{3}{2} $$ Searching via Google did not yield anything useful. I'm wondering if there's some ...
1
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0answers
35 views

How to convert a sum into a definite integral?

I know there is some way to convert a sum into an integral, and vice versa. However, I am very confused on how actually to do this, and if there is some sort of intuition behind it. Note: I'm not ...
2
votes
1answer
25 views

Cute convergence problem. Proving convergence of sequence regarding reciprocals of least common multiple converges.

This is the first problem of the second day of the $2014$ CIIM. Let $\{a_n\}$ be a strictly increasing sequence of positive integers. Prove the sequence ...
2
votes
2answers
71 views

Find the value of: $\sum_{n=1}^{50}n(n!)$

Find the value of the summation: $$\sum_{n=1}^{50}n(n!)$$ The solution in the answer key is $51!-1$. I am unable to find the given solution. Thanks in advance!
0
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1answer
27 views

Discuss whether the series $\sum \left[(\pi/2)^a - (\arctan n)^a\right]$ converges or not, based on the value of $a$

$$\sum_{n=1}^\infty {\left[ {{{\left( {\frac{\pi }{2}} \right)}^a} - {{(\arctan n)}^a}} \right]} $$ I proved that the series diverges for $a < 0 $ and that the series converges for $a = 1$ (using ...
5
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3answers
127 views

Evaluating $\lim\limits_{n\rightarrow \infty} \frac1{n^2}\ln \left( \frac{(n!)^n}{(0!1!2!…n!)^2} \right)$

Evaluating $$\lim\limits_{n\rightarrow \infty} \frac1{n^2}\ln \left( \frac{(n!)^n}{(0!1!2!...n!)^2} \right)$$ I'm not quite sure where to start in evaluating this. Some pointers, or a solution, ...
0
votes
0answers
14 views

Name of the algorithm for an infinite arithmetic series of any number of fractions [on hold]

I was playing with fractions, no number or anything, and I figured out an algorithm for finding the sum of any fractions, even when not in a series so that I can take a number of fractions and create ...
1
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2answers
31 views

Cant find a solution for Dirichlet convolution

I searched for a solution of the sum $$\sum_{d|n}d^2\mu(\frac nd)$$ running over the divisors of $n$ and $\mu$ the Möbius function. In the available textbooks it is not dealt with or I overlooked the ...
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2answers
22 views

Integral of strictly real function has imaginary component

Intuitively and informally speaking, $\int_{a}^{b}f(x)dx$ is summing all of the values $f(x)$ yields for $x\in [a,b]$. So it would make sense that if $f(x)$ is strictly real over $[a,b]$, then ...
3
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3answers
111 views

Proof verification for proving $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ by induction

Prove by mathematical induction: $\forall n \ge 2, 1 + \frac1{2^2} + \frac1{3^2} + \cdots + \frac1{n^2} < 2 − \frac1n$ Basis Step: (We want to show, $P(2)$, which is 1 + ...
0
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0answers
6 views

how I can find sum of elements of array in alternative way?

Suppose I have the following matrix: * a * b * c d e * f * g * H I j * k * L * The first summation will be a+b+...+L I am looking for alternative summation to give me the same ...
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0answers
73 views
+100

Compare $\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ …

Given two integer sequences $\displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ $ B_n=\left\lfloor\dfrac{n^2}{2\varphi}\right\rfloor-\left\lfloor ...
0
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0answers
23 views

Show a function is periodic and find the period

Let $x(t)$ be a continuous signal, and $\hat x(u)$ be the fourier transform of $x(t)$. We define $\sigma_T(u)=\frac{1}{T}\sum_{n=-\infty}^{\infty}\hat x(u-\frac{n}{T})$ Show that $\sigma_T$ is ...
0
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2answers
34 views

Partial derivitive of a summation.

I need some help taking the partial derivative of this function, if it is possible. Thanks!
3
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0answers
35 views

Dealing with non-constant term in Binomial Theorem question

I am wondering this. Suppose I have a sequence $\{\varepsilon_n\}_{n=0}^\infty$ and elements of this sequence are part of a binomial type expression: For example, my expression is ...
0
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0answers
16 views

How do I solve this equation? Limits? K Theory?

How do I solve this problem? K Theory? Let $f \in \mathscr{C}^{(m)}(E),$ where $E$ is an open subset of $R^{n}$. Fix $\textbf{a}$ $\in E$, and suppose $\textbf{x}$ $\in R^{n}$ is so close to ...
3
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1answer
49 views

Finding value of exponential sum

I'd like to find the value of the following sum $$S(u) = \sum_{n=0}^\infty \frac{e^{iu2^n}}{2^{n+1}}$$ for $u \in \mathbb R$, but I can't seem to do it. Unfruitful work Writing $$S = ...
1
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1answer
40 views

How do we approximate sum of random variables?

Suppose we have independent, identically distributed random variables $X_n \notin L^1$. I would like to approximate, in some way, the distribution of their sum $\sum X_n$ .The problem is that these ...
0
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0answers
22 views

Double summation dummy variable change. [duplicate]

I want to demystify my last question that I link here . I think the way I asked it made it unnecessary hard to understand what I wanted Basically it boils down to showing the following equality: ...
0
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1answer
42 views

How to simplify the summation of log

I have a summation that involve log. I don't know how to solve this summation. I want to find an expression (even a good approximation is enough) for this summation. $\sum_{k=0}^{n}{log(a_k)}$ ...
-1
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1answer
33 views

How to find value of i when ∑ from k=1 to i is defined by a recursive formula and equals 982?

Thanks for the pointers! Here's updated and edited question I'm trying to find the number of days it takes to reach 982 miles when you start traveling at 18 miles/day and decrease your speed by 2% ...
6
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4answers
93 views

What does $\sum_{i=1}^{10} 2$ mean exactly?

Suppose I have $$ \large\sum_{i=1}^{10} 2. $$ Do I just add $2$ to itself $10$ times? I have worked on more complex ones with $n$ and such in the place where the $2$ is, but I have never done it when ...
1
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1answer
44 views

Sum limit. Please tell if correct.

I just solved this limit, and the result provided by the book is different. $$ \lim_{x\to 1} \frac{x+x^2+x^3 + ... + x^n - n}{x-1} $$ I turned this into: $$ \lim_{x\to 1} \frac{x-1}{x-1} + ...
1
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0answers
19 views

Calculate the right Riemann sum to approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5]$.

I'm attempting to calculate the right Riemann sum and approximate the area of the region bounded by $f(x) = 25 - x^2$ on the interval $[-5, 5] = [a, b]$. $$\sum_{k = 1}^{n}{f(a + k\Delta x)}\Delta ...
1
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1answer
31 views

On an algebraic manipulation of a double summation used to obtain the kth ordinate of the periodogram.

I am following Introduction to statistical time series by Fuller. I am having some problems with what I think is an algebraic manipulation of the double summations in the line where the mouse pointer ...
0
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0answers
47 views

The formula for a summation for $\sum_{i=0}^n i^n,\,$ for arbitrary $n$? [duplicate]

$$\text{If }\,\sum_{i=0}^n i^2 = \frac{(n^2+n)(2n+1)}{6}$$ $$\text{and }\,\sum_{i=0}^n i^3 = \frac{[n(n+1)]^2}{4},$$ $$\text{is there a formula for }\,\sum_{i=0}^n i^n\;?$$
2
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1answer
24 views

Summation to count number of strings over N characters?

How many different strings of five characters are there if only lower-case letters or numbers can be used in creating these strings? Here is my solution: There are 26 letters in the alphabet ...
2
votes
2answers
22 views

Taylor polynomial manipulation

Find $\sum_{k=1}^\infty\frac{(-1)^{k+1}x^k}{k}$ This is in a section in my book on Taylor polynomials/Taylor series so I assume we have to find some way to manipulate Taylor polynomials to get this. ...
2
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1answer
24 views

Semantic question about summations

This is a simple question about the syntax of summations. Is the following true: $\lambda_1\sum_\limits{j=1}^{\infty} f_1(E_j)+ ...
2
votes
2answers
50 views

How to calculate the sum $\sum_{i=0}^\infty i^nx^i$ [duplicate]

Here I have $x\in\mathbb{R}_+$ and $x < 1$. I would like to evaluate the following sum: $$\sum_{i=0}^\infty i^nx^i.$$ I know that $$\sum_{i=0}^\infty x^i=\dfrac{1}{1-x}.$$ So I started ...
1
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3answers
23 views

Algebraic simplification of likelihood ratio

Can someone help me understand how this: ...
1
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2answers
35 views

Standard result for $\log(x)$

$$\sum_{1\leq m\leq x/d}\frac{1}{m}=\log(\frac{x}{d})+O(1)$$ I read this result in lecture papers I was going through and can't find anything about its origin. Is there a standard summation result ...
1
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2answers
24 views

Mistake in proof of sum of divisors function $\sigma(n)$

The proof derives the correct result, but I cannot see how the first equality is correct. To begin we use the formula $\sigma(n)=\sum_{d\mid n}d$ This is the first step in the proof: $$\sum_{1\leq ...
1
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1answer
32 views

Expected value of Bernoulli with probability of success Gaussian distributed

I have a circle with centre $(0,0)$. I am generating Matlab code to include $N$ neurons in a neural network. The probability of including individual neurons in a network decays exponentially with ...
0
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0answers
10 views

Estimate error in approximating integral with sum

In a paper, for practical purposes the following integral is approximated as a sum, $$ \frac{1}{2\pi} \int_0^{2\pi} f(\theta) e^{-il\theta}d \theta \approx ...
2
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5answers
143 views

Estimate $\int^1_0 e^{-x^2}\, dx$

Estimate $\int^1_0 e^{-x^2}\, dx$ This is in a section on Taylor series so I would assume that is how it should be solved. I started by using the Taylor series formula for $e^x$ replacing $x$ with ...
2
votes
3answers
51 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
0
votes
1answer
42 views

How do we prove $\int \frac{\ln(1+x)}{x}dx = -\sum_{k=1}^{\infty}\frac{(-x)^k}{k^2}$?

After working on the integral $\int_{0}^{1} \frac{\ln(1+x)}{x}dx$ for a couple of hours, I became convinced its antiderivative was not elementary. So I looked it up on Wolfram Alpha, and it found that ...
1
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1answer
54 views

Prove summations are equal

Prove that: $$\sum_{r=1}^{p^n} \frac{p^n}{gcd(p^n,r)} = \sum_{k=0}^{2n} (-1)^k p^{2n-k} = p^{2n} - p^ {2n-1} + p^{2n-2} - ... + p^{2n-2n}$$ I'm not exactly sure how to do this unless I can say: ...