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.
i have to prove that:
n
sum [C(n,k)]^2=C(2n,n)
k=0
i have in my text a hint that i need to use:
(1+x)^n(1+x)^n=(1+x)^2n
but i got that:
n 2n
sum [C(n,k)]^2= sum [C(2n,k)]
k=0 k=0
how do i get out of this mess?
As I don't know how to use this latex coding here it goes...
if I represent by E the sum of terms where k=1 and n is the unknown
I need to use the formulae for Ek and Ek^2 to obtain a formula for Ek(k+1), by simplifying the algebra as much as possible.
Can someone help with this please?
Given U^k_i, the components of U is a delta function i.e for i=k U^i_k =1,
to prove it is invariant under Lorentz transformation~~
I don't know how to express it in Einstein summation notation, I am very confused with the upper-lower index, is it right to write the transformation in this...
Prove that summation of n(n+1)/2 is true for all integers. Why is my proof not valid?
Could someone explain to me how this is not a valid proof of the summation of "i" from i=1 to n:
n(n+1)/2
Show for base cases:
n=1: 1(1+1)/2=1
n=2: 2(2+1)/2=3
n=3: 3(3+1)/2=6
...
inductive...
Hello all,
I am trying to prove that the following is true:
lim_{M \rightarrow \infty} \sum_{P = (\frac{1}{N}-\delta)M}^{(\frac{1}{N}+\delta)M}
\frac{(N-1)^{{M-P}}M!}{P!(M-P)!N^{M}} \rightarrow 1
where P , M , and N are integers, and \delta is an arbitrarily small positive...
2 math questions (summation, mathematical induction)
I have 2 questions regarding summation and mathematical induction
2. Prove by mathematical induction
\sum^n_{r=1} \frac {1}{r(r+2)} = \frac {3}{4} - \frac {(2n+3)}{2(n+1)(n+2)}
i am now trying to prove that
3/4 -...
If \sum^{n}_{r=1} u_r =3n^2 +4n , what is \sum^{n-1}_{r=1}u_r ?
I know that \sum^{n-1}_{r=1}u_r is equals to \sum^{n}_{r=1} u_r =3n^2 +4n - u_n but the answer given is 3n^2-2n-1. How do i express it in that way?
thanks alot.
Help With Probability Question
i have been working on this for a week can anyone help?
NOTE: Look at reply from moodoo for proper matematical symbols!
I need the probability of being dealt a bridge hand with at least 5 hearts.
I have to possible answers but I have never done this...
The set S(N) of all natural numbers is generally believed to have infinite cardinality (ie S(N) has an infinite number of members) and yet every member of the set is believed to be finite. Infinite natural numbers are by convention "not allowed".
This leads to a contradiction, as follows ...
I need to write the following series in summation notation
1) 1+3+5+7+9+11 SUMMAND (2k-1)? is this right?
2) 4+6+8+10+12+12+16+18 (2k+2)? is this right?
Have I got it?
Hey. This has been bugging me for a long time:
why does summation from n=1 to infinity of (-1)^n or i^n or 1/n or -1/n not converge, because summation from n=1 to infinity of 1/n^2 conveges. Don't the terms in (-1)^n or i^n or 1/n or -1/n(-1)^n tend to 0?
From http://mathworld.wolfram.com i see that the integral notation was "the symbol was invented by Leibniz and chosen to be a stylized script "S" to stand for 'summation'. "
So from that i figure integrals are just summations. So what's the difference from Einstein Summation, where "repeated...
I have a rather simple question, but my rusty brain needs a good, swift kick-start.
I start with:
\sum_{i=1}^k i
and substitute in i=k-j to get:
\sum_{k-j=1}^k (k-j)
How do I get from this to the following?
\sum_{k-j=1}^k (k-j) \rightarrow \sum_{j=0}^{k-1} (k-j)
Thanks...
Hello, I really trying to understand what is going on with these summations.
the code is following:
for p = 2 to n
for i = 1 to n - p + 1
j = i + p -1
for k = i to j - 1
O(1) + O(1)
Does j enter anywhere here besides the upper bound of the inner-most...
Rules of Summation...Help Me Please
I have a vauge idea of what these rules mean:
1. \sum^n_{i=1}c=cn
2. \sum^n_{i=1}i=\frac{n(n+1)}_{2}
3. \sum^n_{i=1}i^2=\frac{n(n+1)(2n+1)}_{6}
4. \sum^n_{i=1}i^3=\frac{n^2(n+1)^2}_4
are these rules saying that if i have...
Good evening. I'm having a little difficulty with the summation of rectangular areas when finding the area under a curve.
Question:
Using summation of rectangles, find the area enclosed between the curve y = x^2 + 2x and the x-axis from x=0 to x=3.
Well, I start by dividing the interval...
I've heard something about Poisson summation in relation to Fourier analysis, but I can't seem to find any good info on the subject... Can anyone explain what "Poisson summation" is?
Furthermore, I would like to know exactly what "Parsevals identity" states and how it is applied.
Thanks.
Hi, I don't understand this problem at all:
Rewrite the following sum with the index of summation starting at 3 in summation notation:
\sum_{i=1}^{6}(5+3i)
I know that the sum is 93 but I'm not sure what to do...
Thanks for the help!
Is this simplified?
Use the power rule and the summation rule to find f ' (x) and simplify where possible
f(x) = ((2x^3)/5) - x^2 +3/8
f ' (x) = d/dt(((2x^3)/5) - x^2 +3/8) = ((6x^2)/5) - 2x
Is this the right answer?
How would I solve E1jk without the summation? I know how to solve it using the summation symbol but don't know howto do it without it.
Also, I need help proving that |torque|^2 = |r x F|^2= r^2F^2sin@(thetarF ). r dot F = rF cos (thetarF . Would I have to use (r x F) dot (r x F)?
Summations and calculus gives me fits so please verify my results on these 2 issues:
1. Z = summation ( exp ( - B*E(s)) ) where the sum is over s
d/dB of ln(Z) = d/dB (ln (exp(-BEo) + exp(-BE1) + ... exp(-BEn))
= (exp(-BEo) + exp(-BE1) + ... exp(-BEn))^-1 +
(-E0*exp(-BEo)...