# 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 2017/03/18 17:42 nikolaj 2017/03/18 17:30 nikolaj created 2017/03/18 17:42 nikolaj 2017/03/18 17:30 nikolaj created Line 15: Line 15: $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$ $\frac{\mathrm d}{\mathrm d z}\mathrm{exp}_n(z) = \mathrm{exp}_{n-1}(z) = \mathrm{exp}_n(z) - \dfrac{1}{n!} z^n$ + + == Alternative == + Another series which has $a^z$ as limit is the one given below. It has the nice feature that for integers \$n​ + 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 + ​ === References === === References ===