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.
Homework Statement
I have attached the question, along with the solution in the picture attached. This is one of the few questions I have encountered that I completely have no idea what the solution is trying to do...
It's like they do not make any sense at all!
Confused by
1. The...
Homework Statement
I want to change the order of triple summation. it follows:
$$\sum^N_{k=0} f(k) \sum^k_{n=0}\sum^{N-k}_{m=0} g(k,n)h(k,m)A(n+m)$$
=>
I need to set the variable x(=n+m) go from 0 to N firstly, and then further go on...
$$\sum^N_{x=0} A(x) \cdots \cdots$$
But, I...
Hi, I'm currently taking Discrete Mathematics and I'm working on a mathematical induction problem that's a little different than usual because it has a summation in it. What I basically want to know is did I do parts A and C correctly?
Homework Statement
Homework Equations
The Attempt at a...
can you please tell me what is the basic difference between summation and integration..? i was going through the Poisson distribution function and in one case it was discrete and we had to make summation to get the result and other cases for continuous function we integrated it...now what is...
Homework Statement
Thank you very much for helping me.
I have to convert the following summation of a term from 1 to ∞ to a definite integral.
Sum for k=1 to ∞: (2+Cosh[2k/x]) Csch^4 [k/x]
I have already tried the rules for converting from different sources and websites, which is...
Edit: LOTS OF TYPOS (sorry guys)
Let:
f(r) = e^{-(a-r)^2}
g(r) = r e^{-(a-r)^2}
Where a is some constant
Can:
\dfrac{ \sum\limits^{r=\infty}_{r=-\infty} g(r) } {\sum\limits^{r=\infty}_{r=-\infty} f(r) }
Be simplified?
Hello all,
I'm aware of the Monte Carlo Summation method in discrete spaces, where you can approximate a very long summation over the entire space by a shorter one with only a few randomly selected terms from the original summation (weighted by the inverse probability density of them being...
Hello all,
I'm aware of the Monte Carlo Summation method in discrete spaces, where you can approximate a very long summation over the entire space by a shorter one with only a few randomly selected terms from the original summation (weighted by the inverse probability density of them being...
Is there a general formula for something like
\sum_{n=0}^{\infty} \left( f(n) \times g(n) \right)
For example, what is
\sum_{n=0}^{\infty} \left( 3^n \times \frac{n!}{n^2} \right)
Homework Statement
What is the sum of:
Homework Equations
N/AThe Attempt at a Solution
I'm unsure how to start.
Note: I'm in Grade 10, so I may not have the mathematical skills necessary to understand the solutions you provide.
Any help/guidance would be appreciated.
Homework Statement
Someone in school was showing me this proof or problem that, I believe, proves or yields π via this limit: Lim x-->0 of \frac{xπcot(πx)}{x}-\frac{1}{x} = tan(0) = 0
And that this somehow related to a summation \sum1/k^{2} as the sum goes from 1 to ∞.
I don't...
So basically here's the deal:
I believe there exists a P(x) defined on [-2π, 2π]
such that over that interval P(x) = \sum^{\infty}_{n=0}[sin(πnx)]
Its weird but I have a feeling that this might converge to a function such as tangent
Homework Statement
For formatting sake I've copied a picture of the problem and attached it here: http://i.imgur.com/kOjTy.png
Im not worried about the coding part right now I feel I can handle that, my main issue is trying to understand how the values in the summation are derived. It...
The sum of 1 + 2 + 3...n = n(n+1) / 2 - highest power term is n^2
sum of 1^2 + 2^2 + 3^2...n^2 - n(n+1)(2n+1) / 6 - highest power term is n^3
sum of 1^3 + 2^3 + 3^3...n^3 - it has highest power term of n^4
similarly 1^k +2^k ...n^k - it has highest power term of n^(k+1)
Is it a coincidence...
Show that $\displaystyle \sum_{n=0}^{\infty} (-1)^{n} \arctan \left( \frac{1}{2n+1} \right) = \arctan \Bigg( \text{tanh} \Big( \frac{\pi}{4} \Big) \Bigg)$.I'm tempted to give a hint (or two) right off the bat. But I'll wait.
Homework Statement
http://desmond.imageshack.us/Himg100/scaled.php?server=100&filename=img20120327195119.jpg&res=medium
Homework Equations
The Attempt at a Solution
I just plugged in ∞ for n
[2+\frac{3}{∞}]2 (\frac{3}{∞}) =
[2+0]2 (0) = 0Did I do the problem correctly? I might need a...
Hi
Is it correct of me to say that I want to carry out the sum
\sum_i{v_iw_i}
where i\in\{x,y,z\}? Or is it most correct to say that i=\{x,y,z\}?Niles.
Hello, I am looking at a derivation that involves (note x is a column vector)
\frac {d(\vec{x}^T\vec{x})} {d\vec{x}} = \vec{x}^{T}
So I convert to summation notation and evaluate as follows
\sum_{i,j} \frac {d(x_{i}x^{i})} {dx^{j}}
\sum_{i,j} \frac {dx_{i}} {dx^{j}} x^{i} + \sum_{i,j}...
Let $k$ be a positive integer. Let $n=2^{k-1}$. Prove that, from $2n-1$ positive integers, one can select $n$ integers, such that their sum is divisible by $n$.
Homework Statement
I want to justify that \int_{0}^{1} \frac{f(x)}{1-x} \ dx = \int_{0}^{1} f(x) \sum_{k=0}^{\infty} x^{n} \ dx = \sum_{k=0}^{\infty} \int_{0}^{1} f(x) x^{n} \ dx
Homework Equations
The Attempt at a Solution
I always thought changing the order of summation...
I have the following summation and I'm attempting to remove the summation notation. It appears to be the sum of a geometric series but I'm having a great deal of trouble with it. X is an unknown constant.
$$\sum\limits_{i=2}^n (n - (n-i))x^{n-i}$$
Thanks.
How can I make this mathematically correct? I hope you see what I'm trying to do?...
If you have a graph where:
W=\displaystyle\int^{\theta_{2\pi}}_{\theta_{0}} \tau d\theta
Then the estimated area with the trapesium rule:
\displaystyle\int^{\theta_{2\pi}}_{\theta_{0}} \tau...
Homework Statement
I have trouble with the summation notation.
\sum_{i=0}^{k}\binom{k}{i}f_{n+i}
How do I write this as a sequence based on the definition of Fibonacci sequence?
Homework Equations
Definition:
f(0)=0
f(1)=1
f(n)=f(n-1) + f(n-2) for n>=2
Example:
f(2) = f(1) +...
Determine a formula for the sum of
\sum iri-1
in terms of n and r.
I am stuck on this, i don't understand what to do with the "i" infront of the ri-1i know that \sum ri-1 = (1-rn ) /1-r
all sums are for an index of i=1 to n
I...
I am trying to understand summation notation and there are a few inconsistencies in my head that I would like to clear up.
Suppose C is an m*n matrix and \vec{x} is a 1*m row vector. Then,
\vec{x}C = \sum_{i} x_{i}C_{ij} = \sum_{i} C_{ij}x_{i} = \sum_{i} {C_{ji}}^Tx_{i} = C^T \vec{x}...
I have been working on representing the powers of numbers as a summation.
This is as far as I have gotten.
Power: 2
m^2 = \sum_{n=1}^m \left(2n -1\right)
Power: 3
m^3 = \sum_{n=1}^m \left(3n^2 -3n +1\right)
Power: 4
m^4 = \sum_{n=2}^m \left[6*(4n-6) * \left(\sum_{a=1}^{m-n+1}...
Hi,
I am reading a paper, and at some point the authors claim that:
\sum_{m=1}^{L+1}\frac{\prod_{\substack{l=1\\l\neq m}}^{L+1}\frac{\lambda(m)}{\lambda(m)-\lambda(l)}}{\lambda^r(m)}=0
the question is HOW?
Any tiny hint will be highly appreciated.
Thanks
Homework Statement
Consider any sequence a1, a2,..., an and the nxn array of values bij = aiaj. Which terms in the array are involved in the sums L = Ʃ(between i=1 and n)Ʃ(between j=1 and i) bij
and U = Ʃ(between j=1 and n)Ʃ(between i=1 and j) bij?
Also, by symmetry, show that L=U.
Homework...
Homework Statement
I have the following function,
f(t) = 2048 + 700cos(31.25*2\pi t) - 1100sin(125*2 \pi t)
and I need to find 48 data points spread evenly between one peroid of the waveform.
How would I go about doing this?
Homework Equations
The Attempt at a Solution...
Prove of summation claim ??
Hi every one,,
any idea how to prove the following claim
\sum_{i=0}^{n}a_iz^i=(1-z)^{\binom{m}{2}}(1+z)^{\binom{m+1}{2}}
i think we need to use some derivatives, may be the second derivative will help.
please help.
Hello everybody,
I have some problems in finding an analytical expression for this product:
\sum_{j=0}^{N}(N-j)e^{-ijy}\cdot\sum_{k=0}^{N}(N-k)e^{iky} .
I have solved the problem for several Ns, applying the Euler rule 2\cos(x) = e^{ix} + e^{-ix}
Now, I'm trying to express the...
I have just started on a course in Tensor calculus and I'm absolutely new to it, so I read that according to the summation convention, if an index appears twice, it means that the expression is summed over that index, but if it appears more than twice then the expression is meaningless. I want...
This problem came up in a project I'm doing for work, and I don't have a very extensive math background so I don't know how to solve it. I would appreciate any help you guys could give me.
X1 is a constant
Y1 is a constant
Xn = aXn-1 + bYn-1
Yn = cXn-1 + dYn-1
For all n, for some constants...
Hi. I've finished my undergraduate math methods courses. Many times we had problems where we had a summation and an integral both acting on the same term, and we'd switch the order of the two operations without thinking about it. The professor would always say, "I can interchange these two...
Good day all,
I received this picture through the facebook network last night. I took it as valid at first sight, but for some reason it bothered me; in other words, my skepticism kicked in. The author mentions and assumes the limit as "n goes to infinity" which is annotated in the...
It's been a while since I took Calc 2 and I am in a Linear Systems and Signals class right now. I'm looking at a solution on how to obtain a zero state response of a discrete time signal, but performing the summation confuses me. Can someone explain the steps they did?
This is part A...
I am looking for a bound for the following expression
S=\sum_{n=1}^N n^k e^{-an}
where a>0 and k=1, 2, 3, or 4, apart from the obvious one:
S\le \frac{n+1}{2} \sum_{n=1}^N e^{-an} = \frac{n+1}{2}
\frac{1-e^{-Na}}{e^a-1}
Homework Statement
The Spivak's Calculus Answer Book (3ed) states that, on page 17,
\sum_{i \neq j} (x_{i}^{2}y_{j}^2 - x_{i}y_{i}x_{j}y_{j}) = 2\sum_{i < j}(x_{i}^{2}y_{j}^2 + x_{j}^{2}y_{i}^2 - x_{i}y_{i}x_{j}y_{j})
But as I speculate, I've got the following:
\sum_{i \neq j}...
My logic is flawed somewhere, but I can't figure out where or why.
So I've been playing with summation a bit and figured out a way to make equations for Ʃ^{n}_{k=1}K and Ʃ^{n}_{k=1}K^{2} That looks odd, so I'll just use Ʃ from now on, but realize that it is always from k=1 to n.
ƩK is a series...
Homework Statement
On http://www.chem.arizona.edu/~salzmanr/480b/statt02/statt02.html they're going from a three-fold summation to a single summation (cubed) (equation 58-59), something like this:
\sum_{n_x} \sum_{n_y} \sum_{n_z} e^{-a\left(n_x^2 + n_y^2 + n_z^2\right)} = \left( \sum_n...
Homework Statement
I need some advice on prooving this formula (f is an arbitrary function):
\sum^{N}_{t=1}\sum^{N}_{s=1}f(t-s)=\sum^{N-1}_{τ=-Ν+1}(N-|τ|)f(τ)
Thanks in advance
Does anybody know how to solve the next summation? or process, or mathematical program that can solve this?
[PLAIN]http://img341.imageshack.us/img341/7939/unledkn.jpg
Homework Statement
Show that f(x) = \sum_{i=1}^{\infty}\frac{2^{i}x - \lfloor 2^{i}x \rfloor}{2^{i}} is continuous at all real numbers, excluding integers.
The Attempt at a Solution
I've tried going about via |f(x) - f(y)| < ε, but am having trouble with this, since first, I don't get anywhere...
Homework Statement
A line of buckets numbered 0,1,2... extends indefinitely to the left with an elephant behind each bucket. Initially, all of the buckets are empty but then peanuts start falling into bucket 0 at a rate of one per second for 2^12 seconds. Whenever 5 peanuts accumulate in a...