In mathematics, summation is the addition of a sequence of any kind of numbers, called addends or summands; the result is their sum or total. Beside numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any type of mathematical objects on which an operation denoted "+" is defined.
Summations of infinite sequences are called series. They involve the concept of limit, and are not considered in this article.
The summation of an explicit sequence is denoted as a succession of additions. For example, summation of [1, 2, 4, 2] is denoted 1 + 2 + 4 + 2, and results in 9, that is, 1 + 2 + 4 + 2 = 9. Because addition is associative and commutative, there is no need of parentheses, and the result is the same irrespective of the order of the summands. Summation of a sequence of only one element results in this element itself. Summation of an empty sequence (a sequence with no elements), by convention, results in 0.
Very often, the elements of a sequence are defined, through regular pattern, as a function of their place in the sequence. For simple patterns, summation of long sequences may be represented with most summands replaced by ellipses. For example, summation of the first 100 natural numbers may be written as 1 + 2 + 3 + 4 + ⋯ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where
∑
{\textstyle \sum }
is an enlarged capital Greek letter sigma. For example, the sum of the first n natural integers can be denoted as
∑
i
=
1
n
i
.
{\textstyle \sum _{i=1}^{n}i.}
For long summations, and summations of variable length (defined with ellipses or Σ notation), it is a common problem to find closed-form expressions for the result. For example,
∑
i
=
1
n
i
=
n
(
n
+
1
)
2
.
{\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.}
Although such formulas do not always exist, many summation formulas have been discovered—with some of the most common and elementary ones being listed in the remainder of this article.
Summation Problems! Please Help!
My geography prof assigned these... believe it or not. Its a quiz and its worth 5% of our mark.
1) Σi^4 = 1^xi Variables - n=4 x1=1 x2=6 x3=9 x4=17 (i think its 4)
2) (Σi^4=1^xi)^2 n=4 x1=3 x2=10 x3=9 x4=12 (i think this one is 4 also)
3) Σi^2= 1Σj^2 =...
\sum\frac{1}{(2j-1)^2}
This fgoes from j=1 to infinity. I was just wondering if somebody could calculate and show all working to show the value that this function converges to as i have no idea of how to do this? Thanks for your help
Let k and n \le X be large positive integers, and p is a prime. Define
F(X,n) := \sum_{\substack{k^2+p = n\\X/2\le p<X\\\sqrt{X}/2 \le k < \sqrt{X}}}\log p
Q(n) := \sum_{k^2+p = n}\log p.Note that in Q(n), the ranges of k and p are unrestricted.
My question is:
I know that F(X,n) and Q(n) can...
Homework Statement
Evaluate the following sums, implied according to the Einstein Summation Convention.
\begin{array}{l}
\delta _{ii} = \\
\varepsilon _{12j} \delta _{j3} = \\
\varepsilon _{12k} \delta _{1k} = \\
\varepsilon _{1jj} = \\
\end{array}
The Attempt at a...
Homework Statement
This is kind of a question regarding summation.
All logs are to base 2.
Given
A=\sum_{n=2}^{\infty}(n\log^{2}(n))^{-1}
Why does the the Author get
\sum_{n=2}^{\infty}\frac{\log A}{An\log^{2}(n)}=\log A
?
Homework Equations
The Attempt at a...
\sum_{n=1}^{\infty}n^{-2}=\frac{\pi^2}{6}
I'd like to know how to prove this summation. And if possible, what is the significance of having \pi in the answer?
\sum_{n=1}^{\infty}(-1)^{n}\frac{e^{-\frac{1}{nx}}}{n}
Where 0<x<oo.
I'm looking for a closed form/ closed representation for this series [I was thinking something like a polylogarithm or dirichlet eta function combination might work].
Any ideas or suggestions would be much appreciated.
Can anyone explain this property of shifting the index on the summation notation?
I'm reading a book and came across this which has confused me. I don't see how these are equal:
\sum_{k=1}^n \frac{1}{k(k+1)} = \frac{1}{2} + \sum_{k=2}^{n+1} \frac{1}{k(k-1)}
It's part of an explanation that...
The problem is to prove the following:
\sum_{m>0}J_{j+m}(x)J_{j+m+n}(x) = \frac{x}{2n}\left(J_{j+1}(x)J_{j+n}(x) - J_{j}(x)J_{j+n+1}(x)\right).
Now for the rambling...
I've been reading for a while, but this is my first post. Did a quick search, but I didn't find anything relevant. I could...
This is not really a homework problem, but I'm studying a text, and I came across this:
http://img198.imageshack.us/img198/4586/sumh.jpg
I know how to get that fraction with the exponents in it (using a summation formula). But for the life of me, I can't figure out how to manipulate that...
I need to know if the following series converges:
∑(k=1 to k=oo)[(((-1)^k) ζ(k))/(e^k)]
The problem is that zeta(1)=oo; however, the equation satisfies the conditions of convergence for an alternating series [the limit as k->oo=0 and each term is smaller than the last.]
Any thoughts?
I am trying to understand the derivation of the Poisson's sum formula. Wikipedia's article is like crosswords to me. I checked mathworld's take on it. It looked simple, but it stated that the equation is derived from a more general result without stating or proving that general result. Here's...
Homework Statement
f(x) = \frac{1-cos(X^2)}{x^3}
which identity shoud i use?
and tips on this type of questions? once i can separate them, then i'll be good
thanks!
Hey all,
The way I was taught GR, the summation convention applies on terms where an index is repeated strictly with one covariant, one contravariant. But reading through a translation of Einstein's GR foundations paper just now it looks like the index placement doesn't matter (I've seen it...
Does anyone know if there is a summation formula to find the sum of an expression with n as an argument in a trig function? I'm asking this because I'm learning about Fourier series/analysis but it seems that once we have the Fourier series we only sum for n=1,n=2,n=3... We never sum there...
i need to write this into MATLAB
http://www.engin.umich.edu/class/bme456/ch10fitbiphasic/biphasfit19.gif
which i have done here:
uj = (-sig/Ha)*(xj-(((2*h)/(pi^2))*((((-1)^n)/((n+1/2)^(1/2)))*sin((n+1/2)*((pi*xj)/h))*exp(((-Ha*ko)/(h^2))*((n+1/2)^2)*(pi^2)*t))));
how do i vary n...
Hey,
I have a general question about summations. Is there any steadfast rule for calculating, or obtaining a sometimes-calculatable function for, the derivative of x, where x is the upper bound of summation in a simple summation expression (the summation of f(n), from n = 1 to x)?
If not...
I know how to write it out in the general window, but not in the graphing window (there's no summation option in the graph feature). Is there a way to import it or another way?
Hi,
This is to do with my research. While deriving some theory, I got an equation as follows.
\lim_{n\rightarrow\infty}\sum_{i=1}^n\frac{R^2}{R^2+(4a\,i-2\,k)^2-(4a\,i-2\,k)\,\sin(\gamma)}
Never mind what R, a, k, and \gamma are. They are all constants.
What I would like to do is to get a...
Hey everyone,
I need some help trying to figure out how to find the summation of
n
\sum_{}^{\6}i^p
i=0
I was looking on the web and found on Wikipedia this formula off the http://en.wikipedia.org/wiki/Summation" page. It looks like this assuming I copied it right (ignore the periods)...
Homework Statement
Define I(x)= I( x - x_n ) =
{ 0 , when x < x_n
{ 1, when x >= x_n.
Let f be the monotone function on [0,1] defined by
f(x) = \sum_{n=1}^{\infty} \frac{1}{2^n} I ( x - x_n)
where x_n = \frac {n}{n+1} , n \in \mathbb{N} .
Find \int_0^1 f(x) dx ...
hello guys,
I have tried to evaluate \Sigma e-an2 so many times, but I didn't get it.
where a is just a constant and summation begin from n=1 to infinity.
I know that \Sigmaen is just geometric series which is equal 1/(1-e)
But when n changes to be n2, I have no ideas how to do that.
If...
Fourier series summation...help!
Basically, i need to show that...
2 + sum (m=1 to n) [4(-1)^m . cos(m.pi.x)] = 2(-1)^n.cos((n+1/2)pi.x)/cos((pi.x)/2)
Any ideas?
I have been trying to solve Summation as Limit to Infinity type of questions but there are hardly a few examples I could find in my book
I know the general method for \lim_{n \rightarrow \infty } \frac{1}{n}\Sigma_{r=A(x)}^{B(x)}f\frac{r}{n} where r/n is replaced by x and 1/n by dx, the limits...
I am having trouble understanding how to find the limit of a summation. I know the formulas and properties but cannot seem to simplify them into a rational form becuase i have never been good at simplifying rational expressions and if there is an easier way to solve them.
I enjoy Summation math...
Homework Statement
Let f(n) = 1/2 + 1/3 + ... + 1/n
Show that f(n) is not an integer for any positive integer n
The Attempt at a Solution
I think that rearraning/breaking down the statement might be easier than applying a theorem since it seems like a simpler problem. Simply...
This equation comes out of deriving the canonical partition function for some system. However, the question is more math based. I am having trouble understanding the simplification that was performed in the text:
∑ from N=0 to M of: (M!exp((M-2N)a))/(N!(M-N)!) supposedly becomes...
Homework Statement
\Sigma^{4}_{k=0} \stackrel{1}{k^{2}+1}
Homework Equations
I would imagine it has something to do with this property
\Sigma^{n}_{i=1} i^{2} = \stackrel{n(n+1)(2n+1)}{6}
The Attempt at a Solution
So at first I thought I could bring k^{2}+1 to the top by...
Is there a way to simplify this sum to a generalized function? Would I have to use the gamma function?
\sum^x_{k=0} ({t \choose {2k}}/(2k+1)^y)
where x and y are constants
This is not homework.
I feel so silly asking this question, but is (the summation is over n1 from 1 to infinity. I have no idea how to type it with the latex)
\sum(x1^n1)/n1!*(x^(n-n1))/(n-n1)!
= lim(_{n1 \rightarrow \infty}) (1 + x/n1)^n1 * lim(_{n1 \rightarrow \infty}) (1 + x/(n-n1))^(n-n1)
= exp(x1)*exp(x2)...
Okay, this is a derivation from Relativistic Quantum Mechanics but the question is purely mathematical in nature.
I presume all you guys are familiar with the Levi-Civita symbol. Well I'll just start the derivation. So we are asked to prove that:
[S^2, S_j] =0
Where...
Q1. f is a continuous real valued function on [o,oo) and a is a real number
Prove that the following statement are equivalent;
(i) f(x)--->a, as x--->oo
(ii) for every sequence {x_n} of positive numbers such that x_n --->oo one has that
(1/n)\sum f(x_k)--->a, as n--->oo (the sum is taken...
Hi,
I'm a beginner in C++.
I wan't to write this program:
Write a program that asks the user to enter n numbers –where n entered by the user- and calculates the sum of even numbers only. main function asks the user to enter n and then calls the recursive function Sum to read the values...
[SOLVED] Seperating a Summation problem.
Homework Statement
The Problem:
Separate a sum into 2 pieces (part of a proof problem).
Using: X=
\sum^{n}_{k=1}\frac{n!}{(n-k)!}
Solve in relation to n and X:
\sum^{n+1}_{k=1}\frac{(n+1)!}{(n+1-k)!}
Homework Equations
?
The...
(1/2) + (2/4) + ... + (n/(2^n))
=
sum i=1 to i=infinity of (i/(2^i))?i know how to express the sum of just 1/(2^i), but not the above
thanks for the help!