Rational alternating sequence for exponentiation function?

Question

It seems that rational alternating sequences are useful for generating intervals $[b_1,c_i]$, $[b_2, c_2]$, ... that enclose a value. For example if we take the usual Maclaurin series of sin:

$$\sin(x) = x - \frac{x^3}{3!} + \frac{x^5}{5!} -+...$$ $$= \lim_{n \to ∞} \sum_{i=1}^n (-1)^{i-1} \frac{x^{2i-1}}{(2i-1)!}$$ $$= \lim_{n \to ∞} s_n(x)$$

Then for $x>0$ we can use the following intervals $[b_j,c_j]$ where $b_j=s_{2j}$, $c_j=s_{2j+1}$ which is rational for rational $x$. What would be a known rational alternating sequence for $f(x)=e^x$?

Edit: Note I don't require $b_j$ or $c_j$ constructed as partial sums, it could be also something else, only requirement is that $b_j(x)\le f(x)\le c_j(x)$, and hopefully progressingly smaller.

Answer 1

For $1>x>0$ you may find the following natural: $$\left(1+\frac xn\right)^n<e^x<\left(1-\frac xn\right)^{-n}$$ where both terms converge to $e^x$.

For $x>1$ the inequality should still hold for all $n$ greater than some $N$ that depends on $x$.

Answer 2

I was trying changing the center of the taylor series as follows. If we develop $f(x)=e^x$ at $x=1$ we get the following series:

$$f(x) = e + e (x - 1) + e \frac{1}{2} (x - 1)^2 + ..$$ $$= \lim_{n \to ∞} \sum_{i=0}^n e \frac{1}{i!} (x - 1)^i$$ $$= \lim_{n \to ∞} e\,s_n(x)$$

For values $x < 1$ this series is indeed alternating, in that it produces a sequence of values $e\,s_n(x)$ oscilating around $f(x)$. But the problem is the factor $e$ isn't rational. But since $f(0) = 1$, we can compute $f(x)$ in the following way:

$$f(x) = f(x)/f(0) = (\lim_{n \to ∞} e\,s_n(x)) / (\lim_{n \to ∞} e\,s_n(0))$$ $$= \lim_{n \to ∞} s_n(x)/s_{n+1}(0)$$ $$= \lim_{n \to ∞} t_n(x)$$

Here is an experiment in producing rational intervals for $f(1/2)=\sqrt{e}$ this way. For simplicity I wasn't using rationals but an Excel sheet with floats:

n   an(0)       sn(0)       an(1/2)     sn(1/2)     sn(1/2)/sn+1(0)
0    1.0000000  1.0000000    1.0000000  1.0000000   
1   -1.0000000  0.0000000   -0.5000000  0.5000000   1.0000000
2    0.5000000  0.5000000    0.1250000  0.6250000   1.8750000
3   -0.1666667  0.3333333   -0.0208333  0.6041667   1.6111111
4    0.0416667  0.3750000    0.0026042  0.6067708   1.6548295
5   -0.0083333  0.3666667   -0.0002604  0.6065104   1.6478774
6    0.0013889  0.3680556    0.0000217  0.6065321   1.6488252
7   -0.0001984  0.3678571   -0.0000016  0.6065306   1.6487098
8    0.0000248  0.3678819    0.0000001  0.6065307   1.6487224
9   -0.0000028  0.3678792    0.0000000  0.6065307   1.6487212
10   0.0000003  0.3678795    0.0000000  0.6065307