===== Finite exponential series =====
==== Function ====
| @#55CCEE: context     | @#55CCEE: $ m\in{\mathbb N} |
| @#FF9944: definition  | @#FF9944: $\exp_n: \mathbb C\to\mathbb C$ |
| @#FF9944: definition  | @#FF9944: $\exp_n(z):=\sum_{k=0}^n \dfrac{1}{k!} z^k $ |

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$\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_0(z) = 1 $

$\mathrm{exp}_{n}(z) = \mathrm{exp}_{n-1}(z) + \dfrac{1}{n!} z^n$

== Theorems ==

$\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_0(z) = 0 $

$\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$

== Alternative ==
Another series which has $a^z$ as limit is the one given below. It has the nice feature that for integers $n<m$, it evaluates to $p(m,n)=a^m$ exactly.

<code>
p[m_, z_] := Sum[(a - 1)^k/k! \!\(
\*UnderoverscriptBox[\(\[Product]\), \(j = 0\), \(k - 1\)]\((z - 
       j)\)\), {k, 0, m}] // Expand

Table[Table[p[m, z], {z, 1, m}], {m, 1, 7}]     // 
  Simplify // TableForm

i = 3;
2^i // N
p[5, i] /. {a -> 2} // N

i = 3 + 1/2;
2^i // N
p[5, i] /. {a -> 2} // N
</code>

=== References ===
Wikipedia: 
[[http://en.wikipedia.org/wiki/Exponential_function|Exponential function]], 
[[http://en.wikipedia.org/wiki/Matrix_exponential|Matrix exponential]], 
[[http://en.wikipedia.org/wiki/Exponential_map|Exponential map]]

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=== Context ===
[[Finite sum over a monoid]], 
[[Factorial function]]