Sum (mathematics) | Wikipedia audio article
Channel: wikipedia tts
Date: 2019-10-07

This is an audio version of the Wikipedia Article:
https://en.wikipedia.org/wiki/Summation


00:00:30 1 Notation
00:00:46 1.1 Capital-sigma notation
00:01:17 1.2 Special cases
00:02:18 2 Formal definition
00:02:49 3 Measure theory notation
00:03:20 4 Calculus of finite differences
00:03:51 5 Approximation by definite integrals
00:05:08 6 Identities
00:06:09 6.1 General identities
00:06:40 6.2 Powers and logarithm of arithmetic progressions
00:08:43 6.3 Summation index in exponents
00:10:16 6.4 Binomial coefficients and factorials
00:11:33 6.4.1 Involving the binomial theorem
00:12:04 6.4.2 Involving permutation numbers
00:12:19 6.4.3 Others
00:12:34 6.5 Harmonic numbers
00:12:50 7 Growth rates
00:13:05 8 See also
00:13:21 9 Notes
00:14:22 10 Sources
00:15:09 11 External links
00:15:24 Growth rates
00:16:10 See also
00:16:26 Notes
00:16:41 Sources
00:16:56 External links



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SUMMARY
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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. Besides numbers, other types of values can be summed as well: functions, vectors, matrices, polynomials and, in general, elements of any types 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 does not depend on 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 zero element) results, by convention, 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 1 + 2 + 3 + 4 + ⋅⋅⋅ + 99 + 100. Otherwise, summation is denoted by using Σ notation, where 
  
    
      
        
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    {\displaystyle \textstyle \sum }
   is an enlarged capital Greek letter sigma. For example, the sum of the first n natural integers is denoted 
  
    
      
        
          
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    {\displaystyle \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,

  
    
      
        
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    {\displaystyle \sum _{i=1}^{n}i={\frac {n(n+1)}{2}}.}
  Although such formulas do not always exist, many summation formulas have been discovered. Some of the most common and elementary ones are listed in this article.