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.
So the title pretty much says it all, what other infinite series summations do we have for Pi^2/6 besides,
$$\sum_{n=1}^{\infty} 1/n^2$$
***EDIT*** I should also include,
$$\sum_{n=1}^{\infty} 2(-1)^(n+1)/n^2$$
$$\sum_{n=1}^{\infty} 4/(2n)^2$$
etc. etc.
A unique form outside of the 1/n^2 family.
Hi,
Please see the attached pdf file. Equation 1 and equation 2 are equivalent.
Can someone please help me understand how to simplify equation 1 to get to equation 2?
Thanks.
Hi
I have a textbook which uses the notation:
\sum\limits_{i\neq j}^N a_i a_j
I can't find anywhere what this actually means. Is it equivalent to:
\sum\limits_{i}^N \sum\limits_{j}^N a_i a_j
where j can't equal i?
Thanks.
Homework Statement
Find the sum to n terms of the series whose rth term is 3r - 2r
Homework Equations
The Attempt at a Solution
So I tried this
rlog3 - rlog2 = n(n+1)/2 log3 - n(n+1)/2 log2
then I realized this was kind of useless, the only thing I could get from this is
n(n+1)/2 log...
Hellow!
I want you note this similarity:
\\ \int xdx=\frac{1}{2}x^2+C \\ \int x^2dx=\frac{1}{3}x^3+C
\\ \sum x\Delta x=\frac{1}{2}x^2-\frac{1}{2}x+C \\ \\ \sum x^2\Delta x=\frac{1}{3}x^3-\frac{1}{2}x^2+\frac{1}{6}x+C
Seems there be a connection between the discrete calculus and the...
Homework Statement
Hi
When I want to take the divergence of a rank-2 tensor (matrix), then I have to apply the divergence operator to each column. In other words, I get
\nabla \cdot M = (d_x M_{xx} + d_y M_{yx} + d_zM_{zx}\,\, ,\,\, d_x M_{xy} + d_y M_{yy} + d_zM_{zy}\,\,,\,\, d_x M_{xz} +...
Obviously it makes sense when considering the force of weight and the fact that the object is not moving up or down, but what is it composed of?
What I mean is, the ground you are standing on is made of molecules bonded through the electromagnetic force, right? But those bonds are parallel to...
The Euler-Maclaurin summation formula and the Riemann zeta function
The Euler-Maclaurin summation formula states that if $f(x)$ has $(2p+1)$ continuous derivatives on the interval $[m,n]$ (where $m$ and $n$ are natural numbers), then
$$ \sum_{k=m}^{n-1} f(k) = \int_{m}^{n} f(x) \ dx -...
Hi there, i hope someone can help me.
I am just unsure how to proceed with this problem.
Homework Statement
The question and diagram can be found in the attached image. I am looking for assistance on part (ii)
Homework Equations
1. Summation of Moment = 0
2. Summation of forces along x -...
Question
http://puu.sh/52zAa.png
Attempt
http://puu.sh/52AVq.png
I've attempted to use Riemann sums and use the integral to prove the inequality, not sure if this was the right approach to start with as I am now stuck and don't see what to do next.
For part (b), I know that if (2√n...
Homework Statement
If
$$\csc\frac{\pi}{32}+\csc\frac{\pi}{16}+\csc\frac{\pi}{8}+\csc\frac{\pi}{4}+\csc\frac{\pi}{2}$$
has the value equal to ##\cot\frac{\pi}{A}## then find A.
A)61
B)62
C)63
D)64Homework Equations
The Attempt at a Solution
Writing cosec in terms of sin and taking the LCM to...
Hi,
I am struggling trying to find the (equation of the) pdf of the sum of (what I believe to be) two non-central chi-squared random variables.
The formula given on wikipedia (http://en.wikipedia.org/wiki/Noncentral_chi-squared_distribution) shows that the random variable associated with...
i want to use the Fourier method to fins an summation for
y(x)=Absolute value(sin(x)) -pi<x<pi
i know that cos(pi*n)=(-1)^n
and get why they use cos(pi*(n+1))= (-1)^(n+1)
but why is then cos(pi*(n-1)) also (-1)^(n+1)
(see attachment)
THANX!
Hi PF,
I am working on a parallel reduction code to sum up approximately 1 million 32-bit floating point numbers. The serial part running on each processor uses Kahan summation, no problems there. My problem is that this produces several sum/compensation pairs that now need to be added...
Homework Statement
I'm trying to derive the PGF for the Binomial.
The Attempt at a Solution
I have it whittled down to \sum^{n}_{x=0}(nCx)(\frac{sp}{1-p})^x
I just don't know how to simplify this further. Any help is most appreciated.
Homework Statement
Homework Equations
so i kno the formula for the for the sum of the first N positive integers
when i = 1The Attempt at a Solution
i kno the answer = n^2(n+1)/2
but could someone explain step by step how you reduce it to get the final answer? as if I'm in kindergarten...
Homework Statement
let a_{i}=x^{i} and b_{i}=1\div i ! and c_{i}=(-1)^{i} and suppose that i takes all interger values from 0 to ∞. calculate a_{i}b_{i} and calculate a_{i}c_{i}
Homework Equations
i know that in suffix notation a_{i}b_{i} is the same as the dot product as when you have to...
I've been doing some work with finding the LCM of consecutive integers (1-n) and the number of factors in such an LCM.
It's become easy to construct an LCM and find the factors when n is given.
However, to deconstruct a given LCM and find n is proving difficult.
I eventually would like to find...
Homework Statement
Find a formula for
\sum (2i-1) =1+3+5+...+(2n-1)
Homework Equations
The Attempt at a Solution
\sum(2i-1)=(1+2+3+...+2n)-(2+4+6+...+2n)
=(1+2+3+...+2n)-n(n+1)
I'm unsure what to do with 1+2+3+..+2n ?
Hi all,
I have an all time doubt here. We know that if r.v z = x + y where x and y are 2 random sequences having corresponding pdfs p(x) and p(y), the pdf of z, p(z) = convolution ( p(x),p(y) ). I have seen the derivation for the continuous case although not thorough how to prove it. I...
So i realize that the integral of [f(x)dx] is pretty much the height of the rectangle f(x), multiplied by the width dx. But that is the area of 1 infinitesimally skinny rectangle. How does the integral sign add up an infinite amount of rectangles? I've taken cal 2 so if you could show what the...
M_{K}=\frac{1}{2^{k+1}-2}\sum_{i=0}^{L-1}\sum_{l=1}^{K}\binom{K}{l}h_{i}i^{l}M_{K-l}
M_0=1 and the size of h_i is L.
I tried to compute this summation in matlab, my attempt is as following:
clear
h=[ (1+sqrt(3))/4 (3+sqrt(3))/4 (3-sqrt(3))/4 (1-sqrt(3))/4]';
% for simplicity i take...
Homework Statement
Let ## n \geq 2## be a fixed integer. ##f(x)## be a bounded function defined in ##f:(0,a) \rightarrow R## satisfying
f(x)=\frac{1}{n^2}\sum_{r=0}^{(n-1)a} f\left(\frac{x+r}{n}\right)
then ##f(x)## =
a)-f(x)
b)2f(x)
c)f(2x)
d)nf(x)
Homework Equations
The...
Hello,
I would very much like someone to please clarify the following points concerning tensor summation to me. Suppose the components of a tensor Ai j are A1 2 = A2 1 = A (or, in general, Axy = Ayx = A), whereas all the other components are 0. Is this a symmetrical tensor then? How may Ai j be...
Homework Statement
How can I compute the sum
An example to calculate
\sum_{i=1}^n\sum_{j=i+1}^n(i+2j)?? I only have an example where n=1 and it gives a sum of 0 (why?)
Maybe with n=3, what would the expanded form look like?
Homework Equations
I know how to do double sums, but...
Homework Statement
Find Ʃ(product) with k=1 as the lower limit, and 50 as the upper limit. The formula is k/(k+2)Homework Equations
The Attempt at a Solution
I noticed a pattern where the first few numbers are:
1/3, 2/4, 3/5, 4/6, 5/7 The denominator should cancel with the numerator of the next...
Hii All,
Can anyone give me a hint to evaluate \sum_{n=1}^{\infty}\frac{a^{n}}{n^{1-m}}; Here 0<m,\,a<1.
Please note that the summation converges and < \frac{a}{1-a}.
A tighter upper bound can be achieved as 1+\int_{1}^{\infty}\frac{a^{x}}{x^{1-m}}dx.
Is there any way to get the exact...
Given S, an Infinite Series Summation, find \frac{1728}{485}S
S=1^2+\frac{3^2}{5^2}+\frac{5^2}{5^4}+\frac{7^2}{5^6}+...
I found out the formula for (r+1)th term of the series, hence making the series asS=1+\sum_{r=1}^{\infty}\frac{(2r+1)^2}{(5^r)^2}
Now I have a hard time guessing what to do...
Hello All,
I have what I think an easy summation, but I haven't worked with math for very long - I don't know the term which I should search the internet for in order to solve the problem and so I would be very thankful if you help me get the constant C out of the summation...
I need simplify this equation:
Ʃwixiyi - (ƩxiwiƩyiwi)/Ʃwi
Into an equation of the form: Ʃ(something - something)yi
I am pretty sure the first something is xiwi, but I have no idea what the second something would be...
Any help would be greatly appreciated. Thanks!
Find the sum upto n terms:
1.3.5+3.5.7+5.7.9....tn
I solve it this way:
tn=(2n-1)(2n+1)(2n+3)
Now can I take summation on both sides? How?
I mean when I add 2 on both sides the resultant is 0(2-2=0).Similarly the resultant summation will be zero?
And if I take summation I get...
(Was posted in general physics forum also)
I am currently reading Roger Penrose’s “Road to Reality”. In section 4.3, Convergence of power series, he refers to the sum of the series:
1 + x2 + x4 + x6 + x8 + ... = 1/(1-x2)
Of course, this is true for |x| < 1, beyond which the series...
Hi everybody, I am just trying to find a decent identity that relates the sum $$\sum_{k=0}^{n}a_kb_k$$ to another sum such that ##a_k## and ##b_k## aren't together in the same one. If you don't know what I mean, feel free to ask. If you have an answer, please post it. Thanks in advance!
Homework Statement
I am confused about what happens to the index of summation when I differentiate a series term by term. Let me show you two examples from my diff eq book (boyce and diprima) which are the primary source of my confusion:
Homework Equations
From page 268:
The function f is...
Hi all, I found this "identity" online on Wikipedia, and realized that it would actually come in pretty useful for me, if only I could prove that it is true. Can you guys help me on that?:
$$\sum_{k=1}^nk^m=\frac{1}{m+1}\sum_{k=0}^{m}\binom{m+1}{k}B_k\;n^{m-k+1}$$
where ##B_k## denotes the kth...
I can do this derivation the old fashioned way, but am having trouble doing it with einstein summation notation.
Since \vec{B}=\nabla \times \vec{a}
\vec{B}=\mu_{0}/4\pi (\nabla \times (m \times r)r^{-3}))
4\pi \vec{B}/\mu_{0}=\epsilon_{ijk} \nabla_{j}(\epsilon_{klm} m_{l} r_{m} r^{-3})...
This is a C++ class to be used for summation of doubles (floating point). It uses an array of 2048 doubles indexed by exponent to minimize rounding errors by only adding numbers that have the same exponent.
NUM::NUM - array is cleared out when an instance of NUM is created
NUM::clear() -...
Hi!
I'm currently taking a fairly early stats course, and I'm having a bit of a hangup learning exactly how to use "moments" properly. My general solution whenever I run into problems internalizing things is to do a bunch of easy problems, and to show it from the ground up.
This is my...
Hi,
I have been trying to solve this difficult problem for some time and I thought of at least two ways to prove it but to no avail...the second method that I thought of was to employ binomial expansion on the denominator and that did lead me to the result where it only has x terms in my final...
Homework Statement
Prove the Poisson summation formula.
Homework Equations
The Attempt at a Solution
Correction to image below: the very last line of the theorem (italicized) should say f hat is the Fourier transform, not f(n).
Does this proof make sense and is it...