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− | '''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> |
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+ | :<math>\sum_{k=1}^5k^3=1^3+2^3+3^3+4^3+5^3=1+8+27+64+125=225</math> |
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− | :<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>). |
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⚫ | 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]] |
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− | Infinite sums are divided into convergent and divergent series. With convergent series, the difference between terms decreases (as seen in <math>\sum_{i \mathop =1}^{\infty}\frac{1}{i}</math>), whereas with divergent series, the difference increases (such as <math>\sum_{i \mathop =1}^{\infty}i</math>). The limit of a convergent series is a defined point, whereas the limit of divergent series is infinite. |
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+ | Two convergent series added together with the same index are equal to the series sum of the arguments: |
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− | 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/> |
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+ | Some example sums with closed forms are shown below: |
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− | Sums where <math>i</math> is raised to a power can be found with the following formulas: |
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− | :<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> |
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− | :<math>\sum_{i \mathop =0}^{n}(ar^i)</math>, |
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− | the sum is equal to |
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+ | ==See also== |
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+ | *[[Sequence]] |
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+ | *[[Series]] |
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− | we can use the latter formula. |
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+ | *[[Product operator]] |
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[[Category:Arithmetic]] |
[[Category:Arithmetic]] |
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− | [[Category: |
+ | [[Category:Series]] |
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: