Speed of the usual approximation of the exponential

Asked May 03 '15 at 15:36

Active May 03 '15 at 15:44

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Let's consider the usual approximation of the exponential function $f_n(x)=(1+\frac{x}n)^n$.

What do we know about its speed of convergence to the exponential? That is to say, how can we characterize the functions $g_n(x)=|f_n(x)-e^x|$ and $h_x(n\in\Bbb{N})=|f_n(x)-e^x|$ ?

(Context: I am working on other approximations of the exponential, and I would like to compare those to the usual approximation of the exponential without simply looking at graphs)

edited May 03 '15 at 15:44

asked May 03 '15 at 15:36

Hippalectryon

7,830

It's very slow: you get $f_n(1)=2.7$ for $n=74$, $2.71$ for $n=164$ and $2.718$ for $n=4822$! – Bernard May 03 '15 at 15:56
@Bernard I know, but since I'm looking at $\left(1+\frac{x}{2^n}\right)^{2^n}$ in my research, it converges much faster. – Hippalectryon May 03 '15 at 16:52
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See http://math.stackexchange.com/questions/540982/error-analysis-of-exponential-function?rq=1. – Ted Shifrin May 03 '15 at 18:15

Speed of the usual approximation of the exponential

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