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# Sequences, Sums & Series

In the Wolfram Language, integer sequences are represented by lists.

Use Table to define a simple sequence:

 In[1]:= ⨯ `Table[x^2, {x, 1, 7}]`
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Some well-known sequences are built in:

 In[2]:= ⨯ `Table[Fibonacci[x], {x, 1, 7}]`
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Define a recursive sequence using RecurrenceTable:

(Note the use of {x,min,max} notation.)
 In[1]:= ⨯ `RecurrenceTable[{a[x] == 2 a[x - 1], a[1] == 1}, a, {x, 1, 8}]`
 Out[1]=

Compute the Total of the sequence:

 In[2]:= ⨯ `Total[%]`
 Out[2]=

Compute the Sum of a sequence from its generating function:

 In[1]:= ⨯ `Sum[i (i + 1), {i, 1, 10}]`
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Use ESCsumtESC for a fillable typeset form:

 In[2]:= ⨯ ```\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(10\)]\(i \((i + 1)\)\)\)```
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You can do indefinite and multiple sums:

 In[3]:= ⨯ ```\!\( \*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\( \*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(n\)]i\ j\)\)```
 Out[3]=

Calculate a generating function for a sequence:

 In[1]:= ⨯ `FindSequenceFunction[{2, 4, 6, 8}, n]`
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Generate power series approximations to virtually any combination of built-in functions:

 In[1]:= ⨯ `Series[Exp[x^2], {x, 0, 8}]`
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O[x]9 represents higher-order terms that have been omitted; use Normal to truncate this term:

 In[2]:= ⨯ `Normal[%]`
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Given an unknown or undefined function, Series returns a power series in terms of derivatives:

 In[3]:= ⨯ `Series[2 f[x] - 3, {x, 0, 3}]`
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Convergent series may be automatically simplified:

 In[1]:= ⨯ ```\!\( \*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)] \*SuperscriptBox[\(0.5\), \(n\)]\)```
 Out[1]=

QUICK REFERENCE: Integer Sequences `»`

QUICK REFERENCE: Series Expansions `»`