# Differences

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finite_exponential_series [2017/03/18 17:30] nikolaj created |
finite_exponential_series [2017/03/18 17:42] (current) nikolaj |
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$\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$ | $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$ | ||

+ | |||

+ | == 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 === | === References === |