Editing Series (mathematics) (section)

=== Partial sum of a series ===
Given a series <math display=inline>s=\sum_{k=0}^\infty a_k</math>, its {{tmath|n}}th ''partial sum'' is<ref name=":4" /><ref name=":2" /><ref name=":3" /><ref name=":15" />
<math display=block>s_n = \sum_{k=0}^{n} a_k = a_0 + a_1 + \cdots + a_n .</math>

Some authors directly identify a series with its sequence of partial sums.<ref name=":4" /><ref name=":3" /> Either the sequence of partial sums or the sequence of terms completely characterizes the series, and the sequence of terms can be recovered from the sequence of partial sums by taking the differences between consecutive elements,
<math display=block>a_n = s_{n} - s_{n-1}. </math>

Partial summation of a sequence is an example of a linear [[sequence transformation]], and it is also known as the [[prefix sum]] in [[computer science]]. The inverse transformation for recovering a sequence from its partial sums is the [[finite difference]], another linear sequence transformation. 

Partial sums of series sometimes have simpler closed form expressions, for instance an [[arithmetic series]] has partial sums
<math display=block>
s_n = \sum_{k=0}^{n} \left(a + kd\right)
= a + (a + d) + (a + 2d) + \cdots + (a + nd)
= (n+1)\bigl(a + \tfrac12 n d\bigr),
</math>
and a [[geometric series]] has partial sums<ref name=":45">{{harvnb|Spivak|2008|pp=473–478}}</ref><ref name=":24">{{harvnb|Apostol|1967|pp=388–390, 399-401}}</ref><ref name=":16">{{harvnb|Rudin|1976|p=61}}</ref>
<math display=block>s_n = \sum_{k=0}^{n} ar^k = a + ar + ar^2 + \cdots + ar^n  = a\frac{1 - r^{n+1}}{1 - r}</math>
if {{tmath|r \neq 1}} or simply {{tmath|1= s_n = a(n+1)}} if {{tmath|1= r = 1}}.