(See also section) Tag: sourceedit |
No edit summary |
||
(4 intermediate revisions by 2 users not shown) | |||
Line 1: | Line 1: | ||
− | '''Summation''' is the operation of adding a sequence of numbers to get a sum or total. It is usually denoted with the letter sigma |
+ | '''Summation''' is the operation of adding a sequence of numbers to get a sum or total. It is usually denoted with the letter sigma<math>(\sum)</math>. A sum of all the integers from 1 to 5 can be written as: |
− | :<math>\sum_{ |
+ | :<math>\sum_{k=1}^5k=1+2+3+4+5=15</math> |
⚫ | |||
+ | :<math>\sum_{k=1}^5k^3=1^3+2^3+3^3+4^3+5^3=1+8+27+64+125=225</math> |
||
⚫ | |||
+ | A partial sum, where the sum is only of part of a series, is also called a finite sum. If a sum is between a number and infinity, it is called a [[series]] and is denoted |
||
− | :<math>\sum_{ |
+ | :<math>\lim_{n\to\infty}\sum_{k=1}^n \equiv \sum_{k=1}^{\infty}</math> |
⚫ | Infinite sums can be divergent, meaning they do not converge (such as <math>\lim_{n\to\infty}\sum_{k=1}^nk=+\infty</math> or <math>\sum_{k=1}^{\infty} (-1)^n</math>), or convergent, meaning they equal a specific value (for instance,<math>\lim_{n\to\infty}\sum_{k=1}^n\frac1{k^2}=\frac{\pi^2}{6}</math>). |
||
− | |||
⚫ | |||
==Properties== |
==Properties== |
||
− | A sum of any series in which |
+ | A convergent sum of any series in which <math>f(k)</math> is multiplied by a constant is the same as the entire sum multiplied or divided by said constant. If <math>c</math> is a constant then |
− | :<math>\sum_{ |
+ | :<math>\sum_{k=m}^nc\cdot f(k) =c\cdot\sum_{k=m}^n f(k)</math> |
+ | Two convergent series added together with the same index are equal to the series sum of the arguments: |
||
− | If the sequence is of two numbers added to each other, the answer will be the same as the sums of both terms added together. <br/> |
||
− | :<math>\sum_{ |
+ | :<math>\sum_{k=m}^n(f(k)+g(k))=\sum_{k=m}^n (f(k))+\sum_{k=m}^n (g(k))</math> |
+ | Some example sums with closed forms are shown below: |
||
− | Sums where <math>i</math> is raised to a power can be found with the following formulas: |
||
− | :<math>\sum_{ |
+ | :<math>\sum_{k=1}^nk=\frac{n(n+1)}{2}</math> |
− | :<math>\sum_{ |
+ | :<math>\sum_{k=1}^nk^2=\frac{n(n+1)(2n+1)}{6}</math> |
− | :<math>\sum_{ |
+ | :<math>\sum_{k=1}^nk^3=\left(\frac{n(n+1)}{2}\right)^2=\left(\sum_{k=1}^nk\right)^2</math> |
If a sum is [[geometric series|geometric]], or in the form |
If a sum is [[geometric series|geometric]], or in the form |
||
− | :<math>\sum_{ |
+ | :<math>\sum_{k=0}^nar^k = \frac{a(1-r^{n+1})}{1-r}</math> |
⚫ | |||
− | the sum is equal to |
||
− | :<math>\ |
+ | :<math>\sum_{k=0}^{\infty}ar^k=\frac{a}{1-r}</math> |
⚫ | |||
− | :<math>\lim_{n \to \infty} \frac{a(1-r^{n+1})}{1-r} = \frac{a}{1-r}</math>, |
||
− | we can use the latter formula. |
||
==See also== |
==See also== |
Revision as of 04:09, 12 June 2019
Summation is the operation of adding a sequence of numbers to get a sum or total. It is usually denoted with the letter sigma. A sum of all the integers from 1 to 5 can be written as:
Any operation can be performed on . For instance,
A partial sum, where the sum is only of part of a series, is also called a finite sum. If a sum is between a number and infinity, it is called a series and is denoted
Infinite sums can be divergent, meaning they do not converge (such as or ), or convergent, meaning they equal a specific value (for instance,).
Properties
A convergent sum of any series in which is multiplied by a constant is the same as the entire sum multiplied or divided by said constant. If is a constant then
Two convergent series added together with the same index are equal to the series sum of the arguments:
Some example sums with closed forms are shown below:
If a sum is geometric, or in the form
If the sum of a geometric series is infinite and convergent , the formula simplifies to: