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

learn more… | top users | synonyms

-2
votes
4answers
71 views

Evaluate the sum $1+2+3+…+n$

How do we evaluate the sum: \begin{equation*} 1+2+...+n \end{equation*} I don't need the proof with the mathematical induction, but the technique to evaluate this series.
0
votes
2answers
42 views

Proof of a the following formula: $\sum_{k=0}^n \frac{1}{(k+1)(n-k+1)}=\frac{2}{(n+2)}\sum_{k=1}^{n+1} \frac{1}{k}$

How can I prove the following combinatorial identity about the harmonic series? $$\sum_{k=0}^n \frac{1}{(k+1)(n-k+1)}=\frac{1}{(n+2)}\sum_{k=0}^n \left(\frac{1}{(k+1)} + ...
0
votes
1answer
31 views

Difficulty understanding Sums - Number Theory

I've been trying to understand the following equality for quite some time. And since I bump into it frequently I cannot oversee it: $$\underset{d|n}\sum d=\underset{d|n}\sum ...
0
votes
2answers
9 views

Given the integral of an equation over one set of bounds find the integral over another set of bounds.

If $\int_{1}^{3}f(w)dw=7$, find the value of $\int_{1}^{2}f(5-2x)dx=7$ I think this problem has something to do with the fact that (5-2(2)) = 1 and (5-2(1)) = 3 and these are the bound of the ...
3
votes
5answers
54 views

Formula for $r+2r^2+3r^3+…+nr^n$ [duplicate]

Is there a formula to get $r+2r^2+3r^3+\dots+nr^n$ provided that $|r|<1$? This seems like the geometric "sum" $r+r^2+\dots+r^n$ so I guess that we have to use some kind of trick to get it, but I ...
0
votes
0answers
9 views

recurrence tree final step - binary search

Starting with the base case and recursive case run times as follows: 􏰀 t(N) = 1 , if N = 1 t(N)= 1+t(N/2) ,ifN > 1 At the end of my tree I have ...
2
votes
2answers
55 views

How to obtain a closed form for summation over polynomial ($\sum_{x=1}^n x^m$)? [duplicate]

What is the method for obtaining the polynomial equal to \begin{equation*} \sum^{n}_{x=1}x^m \end{equation*} for unknown $n$, and systematically for various values of $m$? I know it should be a ...
0
votes
0answers
27 views

“+” operator placed as index

What is the meaning of $(a-b)_+$? In other words, what is meant by the "+" operator when it is placed as an index. If I am comparing for example two variables $a$ and $b$. So what is the value of ...
0
votes
0answers
36 views

Is this function $2\pi$-periodic?

If I construct $$G(x) =\sum_{1}^{\infty} g(x +2n\pi),$$ does this make $G(x)$ $2\pi$-periodic? My understanding is that if $G(x)$ were now $2\pi$-periodic, then that means $G(x) = G(x + 2\pi$) = G(x ...
-1
votes
0answers
34 views

A math problem I don't really understand infinitely repeating 0.99999 [duplicate]

Lets say x = 9.999...[repeating] so 10x=9.999... 10x-x=9.999...-0.999 9x=9 so x=1 A friend of mine just posted this on facebook, anyone have a good response?
2
votes
3answers
53 views

Differentiation method for evaluating $ \sum_{n=1}^\infty \frac{n^2}{3^n} $

I evaluated the following infinite sum (the original and broader question regarding this sum can be found at Evaluating $\sum_{n=1}^\infty \frac{n^2}{3^n} $). $$ \sum_{n=1}^\infty \frac{n^2}{3^n} $$ ...
0
votes
0answers
33 views

How to convert infinite intergral to sum

How to convert Wiener filter formulas from integral to sum? They are for images therefore it must be possible to convert them to sums. Any help will be appreciated: I could not find much info on ...
2
votes
1answer
26 views

Why is $\sum_{x=1}^{p-1}\left(\frac{x}{p}\right)=\left(\frac{0}{p}\right)$?

For $p$ an odd prime, why does $$\sum_{x=1}^{p-1}\left(\frac{x}{p}\right)=\left(\frac{0}{p}\right)$$ where $\left(\frac{x}{p}\right)$ is the Legendre symbol. I'm not sure if I have given enough ...
0
votes
3answers
65 views

What does $\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} $ simplify to?

A solution to one of the exercises in my text states: $$\sum\limits_{n=1}^{N-1} \frac{1}{n} - \sum_{n=3}^{N+1} \frac{1}{n} = \frac{1}{1} + \frac{1}{2} - \frac{1}{N} - \frac{1}{N+1}$$ I have no idea ...
1
vote
2answers
69 views

Confusion about how to prove $\sum_{i=0}^n 2^i = 2^{n+1}-1$ for all $n\geq 0$ by induction

I'm trying to understanding proof by induction. But how do I check if that is correct? How do I know what I need to show? Any help would be great. Just trying to get my head around this. So I have ...
1
vote
2answers
60 views

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

My Precalc teacher gave me this as a question and I simply cannot figure out how to do it.
1
vote
1answer
23 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
31 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
25 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
36 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
2answers
97 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\ ...
1
vote
1answer
41 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
votes
1answer
22 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
61 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
votes
4answers
121 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
vote
0answers
37 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
2answers
34 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
73 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
votes
1answer
29 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
votes
3answers
145 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, ...
1
vote
2answers
47 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 ...
0
votes
2answers
25 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
votes
3answers
116 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
votes
0answers
7 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 ...
10
votes
1answer
220 views

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

Given two integer sequences \begin{equation*} \displaystyle A_n=\sum_{k=1}^n \left\lfloor \frac{k}{\varphi}\right\rfloor$ \end{equation*} \begin{equation*} ...
0
votes
0answers
25 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
votes
2answers
40 views

Partial derivitive of a summation.

I need some help taking the partial derivative of this function, if it is possible. Thanks!
3
votes
1answer
63 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
votes
0answers
18 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
votes
1answer
56 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
vote
1answer
45 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
votes
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
votes
1answer
45 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
votes
1answer
34 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
votes
4answers
99 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
vote
1answer
46 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
vote
0answers
20 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
vote
1answer
33 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
votes
0answers
49 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
votes
1answer
27 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 ...