Wolfram Language Fast Introduction for Math Students
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Sequences, Sums & Series

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

Use Table to define a simple sequence:

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

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

(Note the use of {x,min,max} notation.)
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RecurrenceTable[{a[x] == 2 a[x - 1], a[1] == 1}, a, {x, 1, 8}]
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Compute the Total of the sequence:

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⨯ 
Total[%]
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Compute the Sum of a sequence from its generating function:

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

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

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⨯ 
\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(i = 1\), \(n\)]\(
\*UnderoverscriptBox[\(\[Sum]\), \(j = 1\), \(n\)]i\ j\)\)
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Calculate a generating function for a sequence:

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

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⨯ 
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:

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

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

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⨯ 
\!\(
\*UnderoverscriptBox[\(\[Sum]\), \(n = 0\), \(\[Infinity]\)]
\*SuperscriptBox[\(0.5\), \(n\)]\)
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QUICK REFERENCE: Integer Sequences »

QUICK REFERENCE: Series Expansions »