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Let $c>1,k\in\mathbb{N}$.

Let's consider two approximations of the exponential function :

The first one is the most common one $f_k(x)=\left(1+\frac{x}{c^k}\right)^{c^k}$

and the second one is $q_k(x)=\displaystyle\sum _{p=0}^k\frac{c^{\frac{p^2+p}{2}}}{\displaystyle\prod _{n=1}^p(c^n-1)\prod_{n=1}^{k-p}(1-c^n)}f_p\left(x\right)$.

It has been shown here that $q_k(x)$ goes to $e^x$ as $c\to\infty$ or $k\to\infty$.

In the following question, we'll stay with with fixed $c,k$

However, after experimenting a bit on desmos, I have conjectured some results I am not able to prove mathematically, and therefore have some question :

One can see (on the desmos graph) that there exists a non empty interval $I$ around $0$ on which $q_k(x)$ is a better approximation of $e^x$ than $f_k(x)$ (by that I mean that $\forall x\in I,|q_k(x)-e^x|\le|f_k(x)-e^x|$). As $c$ or $k$ increase, this interval gets larger and larger.

  • How can one prove that such an interval exists ?

  • What are the bounds of this interval ?

  • It seems to me that on this interval, $q_k(x)$ 'sticks' to $e^x$ way closer than $f_k(x)$ does. I conjectured that $\forall\epsilon,c>0,\exists k\in\mathbb{N},\forall x\in[-c,c],\left|\dfrac{q_k(x)-e^x}{f_k(x)-e^x}\right|<\epsilon$. How can one show that ?

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Hippalectryon
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1 Answers1

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Well, you've noted that the two functions work best when $c\to\infty$ or $k\to\infty$. So obviously, as $c,k$ increases, the approximations get better. Moveover, it appears $c,k$ can be small as long as $x$ is small. As $c,k$ increases, the range of values $x$ can take and still be reasonably accurate increases. This is true with most approximations.

Also note that your $f_k(x)$ is a really bad approximation if $k$ is small. For that function to be close to $e^x$, $k$ needs to be arbitrarily large.

$c$, on the other hand, can be somewhat smaller because $g_c(x)$ is a better approximation. It just simply is.

In fact, one proves $\lim_{k\to\infty}f_k(x)=e^x$ using calculus. But inputting any more finite number in for $k$ produces exponential functions that are obviously wrong.

In general, one does not use $f_k(x)$ because approximations like $g_c(x)$ are much better.

I've no answer for the exact boundaries of this interval though.

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