What is Sigma notation: Definition and 60 Discussions
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.
okay so I'm a Electrician I've found short method of calculating the final magnitude of a system (Lₜ), this relies mainly on Eucliud's axiom of angles within parrallel lines and is this
∑(cos(θₙ-θₜ)⋅Lₙ)=Lₜ
where Lₙ and θₙ are the initail manignitude and angles respectively, and θₜ is the final...
Hi, I'm currently a Grade 11 student and I need help for this question (Precalculus):
If $\sum\limits_{i=1}^{50} f(i)=90$ and $\sum\limits_{i=30}^{50} g(i)=60$, what is the value of $\sum\limits_{i=1}^{50} (7 g(i)-f(i)+12)/(2)$?
P.S. To those who could answer this, it would be a great help...
Mentor note: Moved from a technical section, so is missing the homework template.
Hi,
I'm always not sure how to prove something in math and I'm wondering if this is enough.
##\vec r \cdot (\vec u + \vec v) ##
##\vec u + \vec v = (u_1+v_1, u_2+v_2,u_3+v_3) = \vec s##
##\vec r \cdot (\vec u +...
Homework Statement
The question involves using sigma notation of Riemann sums to find the area under the graph of ##x^2+\sqrt {1+2x}##. I managed to calculate most of the values and I have ##16+\frac 8 3 + \Sigma {\frac 2 n \sqrt {9 + \frac {4i} n}}##
Homework Equations [/B]
##\Sigma i= \frac...
Prove by mathematical induction that
n
sigma r^3 = n^2(n+1)^2/4
r = 1
so far I have
1
sigma r^3 = 1^2(1+1)^2/2
r=1
1 = 1(4)/2
1 = 4/2
1 = 2
I'm not sure what to do after this for the k+1 case.
Hello everyone!
I have this expression which I have to simplify:
∑i=1log(n) 1(log(n) - i)
And my book apparently simplifies it to being log(n).
I am struggling to figure out why this is the case. Could anyone help?
Thanks!
Homework Statement
Homework EquationsThe Attempt at a Solution
I tried to rationalize the fractions by multiplied it by $$\sqrt{n + 1} - \sqrt{n} $$
it will be sum of $$ \frac {\sqrt{n + 1} - \sqrt{n}} {2n +1} $$
also tried to see the arithmetic structure
$$ 1 + \frac{1}{\sqrt{2} +...
Homework Statement
A) use sigma notation and calculate sum as far as you wish (set your own endpoint)(1/2) + (1/4) + (1/8) + (1/16)...B) this is a personal problem because I want to know how to calculate the sums with e.g. sigma notation, with a calculator.
Never having done this before...
Homework Statement
http://postimage.org/][/PLAIN] [Broken]
image url
Homework Equations
N/A
The Attempt at a Solution
I have to solve a different question using a similar method to this one but I cannot figure out how they got the sigma notation into the last equation format. I tried writing...
Homework Statement
I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.
Homework EquationsThe Attempt at a Solution
Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this...
Hello everyone!
I have this polynomial: $p(x) =$ 1 + \sum_{k=1}^{13}\frac{(-1)^k}{k^2}x^k
- I'm supposed to show that this polynomial must have at least one positive real root.
- I'm supposed to show that this polynomial has no negative real roots.
- And I'm supposed to show that if $z$ is...
Homework Statement
Notice that ln [∏(k=1)^n a^k] = Ʃ_(k=1)^n * ln (a_k)
I couldn't get the LaTeX right on this ^ But k=1 is below the product sign, and n is above. And (a^k) is the formula.
From this, as well as some calculus, calculate that:
lim as n->∞ ∏_(k=1)^n e^\frac{k^2}{n^3}
For...
Homework Statement
a_{1}r^{1} + a_{1}r^{2} + a_{1}r^{3} +... + a_{1}r^{n-3} + a_{1}r^{n-2} + a_{1}r^{n-1} = \sum_{n=1}^{?} a_{1}r^{n-1}
What value would replace the "?"
2. The attempt at a solution
EDIT: I edited this post after receiving your reply Cepheid.
My gut would say...
Here are the questions:
Here is a link to the questions:
How do you write the summation notation for: 4 - 24 + 144 - 864 + ... ? - Yahoo! Answers
I have posted a link there so the OP can find my response.
Homework Statement
My Calc II final is tomorrow, and although we never learned it, it's on the review.
So I have a few examples. Some I can figure out, some I cant.
Examples: f(x)=sinh(x), f(x)=ln(x+1) with x0=0, f(x)=sin(x) with x0=0, f(x)=1/(x-1) with x0=4
The only one of those that I was...
How do I evaluate the sum of this sigma notation problem?
20
∑ k
k=10
Normally, I would think to use the theorem for the sum of the first n integers:
n
∑ k = n(n+1)/2
k=1
I don't know how to do this, however, since the lower limit is k=10, not k=1.
My professor wrote this note on the board...
Hi, I've been wondering this since I started learning integration. I get that ∫ is basically an elongated S for "sum", because that is what it is basically doing. But then Ʃ does the same thing as well. If I'm understanding the difference, it is that Ʃ increments by finite measures, whereas ∫...
Hello,
I have read several different sources on this very topic, and the one thing that confused a little was defining it using sigma notation. Could some please explain to be what it means?
Homework Statement
Prove that:
\sum_{n=1}^{14} 10n = \sum_{n=1}^{7} (20n+70)
Homework Equations
properties of sigma notation
The Attempt at a Solution
I know several properties of sigma notation but none that I know can be used to prove this. I don't know how to change 10 n to 20...
I was playing a bit with the Riemann Zeta function, and have been struggling with some notation problems.
The function is defined as follows
\zeta (s) = \sum_{n=1}^{\infty} \frac{1}{n^s}
where s \in \mathbb{C}
we know that
n^s = exp(s\;ln\;n)
so I can write
\zeta (s) =...
Homework Statement
Taken from Wikipedia's Palindromic number entry, under formal definition header:
Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural numbers in any numeral system. Consider a number n >...
Homework Statement
I'm actually asked to calculate the area under the curve 5x + x2 over the interval [0,1] using Riemann Sums. I found the formula for the Riemann sum over the interval, it being the following:
\sum^{n}_{k=1} (\frac{5k}{n}+\frac{k^{2}}{n^{2}})(\frac{1}{n})
However, I...
Homework Statement
"Express the following sum in sigma notation:
1 + (2/3) + (3/5) + (4/7) + (5/9)"
Homework Equations
The Attempt at a Solution
I've figured out what they all have in common (1+2=3, 2+3=5, 3+4=7, 4+5=9) but I've been searching through the book and on the...
Homework Statement
2n-1
Sigma (3i+1) = n(6n-1)
i=0
prove for all positive n
Homework Equations
The Attempt at a Solution
It holds true for n=1
5=5
then P: m+1
2m+1
Sigma(3i+1) = (m+1)(6(m+1)-1) or 6m^2 + 11m + 5
i=0
then 2m+1
Sigma(3i+1) = m(6m-1) +...
Homework Statement
a) nƩr2(r-1)r=1
Homework Equations
Using the summation series formulae...
The Attempt at a Solution
So far I have got:
r2(r-1) = r3-r2
Ʃr3 = \frac{1}{4}n2 (n+1)2
Ʃr2 = \frac{1}{6}n(n+1)(2n+1)
Therefore,
Ʃr3-r2 = \frac{1}{4}n2 (n+1)2 -...
Homework Statement
In each case, x is an integer between -6 and 6 inclusive.
Homework Equations
x
Σ 2i =12
i=1
The Attempt at a Solution
2x1 = 12 + 2x2 = 12 +...+2x(x) = 12
What does this mean? (see attachment)
"Where b is a block defined by the contiguity condition c that may exist between elements of s, and n is the number of elements in that block"
I know is not possible to get a solution without the actual function, but how does this reads?
Converting Sigma notation...
Homework Statement
(Not a homework question)
Hi!
I have been encountering problems in Binomial Theorem which includes converting the sigma notation.
Like \sum_{k=0}^n \frac{n!}{(n-k)!k!} a^kb^{n-k}=(a+b)^n
I got many questions in my exam of this type with four...
Express this in sigma notation?
3^3 - 3^4 + 3^5 - ... - 3^100
Evaluate these two sigmas?
n
∑ (i-2)^2
i =1
n
∑ (4-i^2)
i =1
I don't really understand sigma notation so I'm really interested in the process and explanation of how to do it. Any help would be greatly appreciated!
Homework Statement
1/1(4) + 1/4(7) + 1/7(10)+...+ 1/(3n-2)(3n+1)
and the sigma notation is pretty obvious
Homework Equations
nothing really..
The Attempt at a Solution
I can see an obvious pattern its n(n+3) but n cannot be 2,3,5,6 etc.. and the second digit in the first term...
The problem posed is:
Evaluate
\sum_{i=1}^{n} (2i + 2^i)
I know that I can break the summation down to this:
2\sum_{i=1}^{n} i\, +\, 2\sum_{i=1}^{n}1^i
and then after using some Fundamental Theorems...
=2\Bigg(\frac{n(n+1)}{2}\Bigg) + 2^n
I can't seem to get it to look like my answer...
how would i plug this into the calculator? to find the sum?
step by step process? thankss!
4
∑ 2 cos (π/k)
k = 1
i tried this site
http://mathbits.com/mathbits/tisection/Algebra2/summation.htm [Broken]
and i got a different answer...so i just need help with the set up on a T.I calc.
I've been clawing at my mind for a while regarding this, I pray somebody can help me.
I was implementing a method from a computer science paper when I came across this:
[PLAIN]http://www.mattkent.eu/challenge.png [Broken]
Bearing in mind that x is a vector, epsilon is a vector of...
A vector A, can be expressed in sigma notation as
\sum Ai where i runs from 1 to 3, i.e. A1 for x coordinate, A2 for y coordinate and A3 for z coordinate.
I wonder how to express vector A in polar form using sigma notation. Could anyone share their knowledge to me?
1. Write each series using sigma notation.
2. The series is:
2 + 5 + 10 + 17 + 26 + 37
3. I know that above the sigma, the number is 6 because there are only six terms in the series. The trouble I seem to have with this is the fact that the series is neither arithmetic or geometric. It...
I have a question about sigma notation and parentheses. Does
\sum x[k] + x[n]
mean
\sum( x[k] + x[n]) or \left(\sum x[k] \right)+ x[n]
I find it a personal annoyance for things to be written with parentheses even if there are rules about it. Including parentheses would just reduce...
1. The problem statement, all variables and given/nknown data
Find a formula for the expression. Use sigma notation.
(x + 1/y)n
The Attempt at a Solution
Not sure how to do this we have only dealt with sigma notation involving only n as a variable
I understand how to solve sigma notation problems where the index variable is equal to 1, but how would I solve a problem like this??
n
\Sigma (i2-3)
i=3
If i was equal to 1 I would be able to solve this, but I'm not sure what to do since it is equal to 3.
Thanks!
Homework Statement
Evaluate the telescoping sum.
a) Sigma i=1 to n [i^4 - (i - 1)^4]
Homework Equations
sigma i=1 to n [f(1) - f(i+1)]
The Attempt at a Solution
so i plug in for the eqns. and get 1^4 - n^4
the correct answer should be n^4 but don't know how to get that.
Homework Statement
Prove the magnitude R of the position vector \vec{R} for the center of mass from an arbitrary point of origin is given by the equation
M^{2}R^{2} = M\sum{m_{i}r^{2}_{i} - \frac{1}{2}\sum{m_{i}m_{j}r_{ij}^{2}
Homework Equations
\vec{R} = \frac{1}{M}...
[SOLVED] Sigma Notation Question/Trig Identity
I posted this elsewhere but I think I put it in the wrong place so I'm going to post my question again here.
Basically I have to deduce the second formula from the first. Both equations are the same except for the top of the right side...
I have no idea how to use the forum equation code so I just attached the question as a word document. I have no idea how to start this one. Any help is appreciated.
Edit: Ok I just took a screenshot and uploaded it as an image.
[SOLVED] Sigma notation of a series.
I have the formula
1+2+3+...+n = (n^2+n+1)/2,
and I thinkthat this is the formula for the sum of a series. I need to write this thing in sigma notation, and then prove it by induction. I'm usually good and proving things by induction, but I can't...