limit((1 + 1/x)^x, x=oo)

488 days ago by mpaul

Let's find \lim_{x \rightarrow \infty}(1 + \frac{1}{x})^x.
limit((1+1/x)^x, x=oo) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}e
\newcommand{\Bold}[1]{\mathbf{#1}}e
e.n(digits = 100) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427
\newcommand{\Bold}[1]{\mathbf{#1}}2.718281828459045235360287471352662497757247093699959574966967627724076630353547594571382178525166427

It would be nice to see why \lim_{x \rightarrow \infty}(1 + \frac{1}{x})^x = e.

How could we expand (1 + \frac{1}{x})^x?

Sage lets us expand binomials very easily:
var('a b') expand((a-b)^10) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}a^{10} - 10 \, a^{9} b + 45 \, a^{8} b^{2} - 120 \, a^{7} b^{3} + 210 \, a^{6} b^{4} - 252 \, a^{5} b^{5} + 210 \, a^{4} b^{6} - 120 \, a^{3} b^{7} + 45 \, a^{2} b^{8} - 10 \, a b^{9} + b^{10}
\newcommand{\Bold}[1]{\mathbf{#1}}a^{10} - 10 \, a^{9} b + 45 \, a^{8} b^{2} - 120 \, a^{7} b^{3} + 210 \, a^{6} b^{4} - 252 \, a^{5} b^{5} + 210 \, a^{4} b^{6} - 120 \, a^{3} b^{7} + 45 \, a^{2} b^{8} - 10 \, a b^{9} + b^{10}

But it doesn't expand (1 + \frac{1}{x})^x any further than it already is:
var('x') (1+1/x)^x 
        
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\frac{1}{x} + 1\right)}^{x}
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\frac{1}{x} + 1\right)}^{x}
expand((1+1/x)^x) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\frac{1}{x} + 1\right)}^{x}
\newcommand{\Bold}[1]{\mathbf{#1}}{\left(\frac{1}{x} + 1\right)}^{x}

So let's create a generator that will expand a power of a binomial using Newton's method:
def newton(a,b,n): C,r = 1,0 while C != 0: yield C*a^(n-r)*b^r C *= (n-r)/(r+1) r+=1 
       
var('x') t = newton(1, 1/x, x) 
       
terms = [next(t) for i in range(18)] terms 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, \frac{x - 1}{2 \, x}, \frac{{\left(x - 2\right)} {\left(x - 1\right)}}{6 \, x^{2}}, \frac{{\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{24 \, x^{3}}, \frac{{\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{120 \, x^{4}}, \frac{{\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{720 \, x^{5}}, \frac{{\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{5040 \, x^{6}}, \frac{{\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{40320 \, x^{7}}, \frac{{\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{362880 \, x^{8}}, \frac{{\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{3628800 \, x^{9}}, \frac{{\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{39916800 \, x^{10}}, \frac{{\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{479001600 \, x^{11}}, \frac{{\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{6227020800 \, x^{12}}, \frac{{\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{87178291200 \, x^{13}}, \frac{{\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{1307674368000 \, x^{14}}, \frac{{\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{20922789888000 \, x^{15}}, \frac{{\left(x - 16\right)} {\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{355687428096000 \, x^{16}}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, \frac{x - 1}{2 \, x}, \frac{{\left(x - 2\right)} {\left(x - 1\right)}}{6 \, x^{2}}, \frac{{\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{24 \, x^{3}}, \frac{{\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{120 \, x^{4}}, \frac{{\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{720 \, x^{5}}, \frac{{\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{5040 \, x^{6}}, \frac{{\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{40320 \, x^{7}}, \frac{{\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{362880 \, x^{8}}, \frac{{\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{3628800 \, x^{9}}, \frac{{\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{39916800 \, x^{10}}, \frac{{\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{479001600 \, x^{11}}, \frac{{\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{6227020800 \, x^{12}}, \frac{{\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{87178291200 \, x^{13}}, \frac{{\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{1307674368000 \, x^{14}}, \frac{{\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{20922789888000 \, x^{15}}, \frac{{\left(x - 16\right)} {\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{355687428096000 \, x^{16}}\right]

Now let's take the limit of each term individually:
[limit(term, x=oo) for term in terms] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \frac{1}{720}, \frac{1}{5040}, \frac{1}{40320}, \frac{1}{362880}, \frac{1}{3628800}, \frac{1}{39916800}, \frac{1}{479001600}, \frac{1}{6227020800}, \frac{1}{87178291200}, \frac{1}{1307674368000}, \frac{1}{20922789888000}, \frac{1}{355687428096000}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \frac{1}{720}, \frac{1}{5040}, \frac{1}{40320}, \frac{1}{362880}, \frac{1}{3628800}, \frac{1}{39916800}, \frac{1}{479001600}, \frac{1}{6227020800}, \frac{1}{87178291200}, \frac{1}{1307674368000}, \frac{1}{20922789888000}, \frac{1}{355687428096000}\right]
sum(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{7437374403113}{2736057139200}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{7437374403113}{2736057139200}
n(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}2.71828182845905
\newcommand{\Bold}[1]{\mathbf{#1}}2.71828182845905

Fascinating thing - the terms above are the same as the terms of \sum_{x=0}^{\infty}\frac{1}{x!}:
[1/factorial(x) for x in range(18)] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \frac{1}{720}, \frac{1}{5040}, \frac{1}{40320}, \frac{1}{362880}, \frac{1}{3628800}, \frac{1}{39916800}, \frac{1}{479001600}, \frac{1}{6227020800}, \frac{1}{87178291200}, \frac{1}{1307674368000}, \frac{1}{20922789888000}, \frac{1}{355687428096000}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, 1, \frac{1}{2}, \frac{1}{6}, \frac{1}{24}, \frac{1}{120}, \frac{1}{720}, \frac{1}{5040}, \frac{1}{40320}, \frac{1}{362880}, \frac{1}{3628800}, \frac{1}{39916800}, \frac{1}{479001600}, \frac{1}{6227020800}, \frac{1}{87178291200}, \frac{1}{1307674368000}, \frac{1}{20922789888000}, \frac{1}{355687428096000}\right]
sum(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{7437374403113}{2736057139200}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{7437374403113}{2736057139200}
n(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}2.71828182845905
\newcommand{\Bold}[1]{\mathbf{#1}}2.71828182845905

So it appears that \lim_{x \rightarrow \infty}(1 + \frac{1}{x})^x = \frac{1}{0!} + \frac{1}{1!} + \frac{1}{2!} + \frac{1}{3!} + ... = \sum_{x=0}^\infty \frac{1}{x!}=e
 
       

 Now here's something weird: 

\lim_{x \rightarrow \infty} (1 - \frac{1}{x})^x = \frac{1}{e}

Isn't that a little surprising?  Check it out -
var('x') limit((1-1/x)^x,x=oo) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-1\right)}
\newcommand{\Bold}[1]{\mathbf{#1}}e^{\left(-1\right)}
n(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
n(1/e) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
t = newton(1, -1/x, x) 
       
terms =[next(t) for i in range(18)] terms 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, -1, \frac{x - 1}{2 \, x}, -\frac{{\left(x - 2\right)} {\left(x - 1\right)}}{6 \, x^{2}}, \frac{{\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{24 \, x^{3}}, -\frac{{\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{120 \, x^{4}}, \frac{{\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{720 \, x^{5}}, -\frac{{\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{5040 \, x^{6}}, \frac{{\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{40320 \, x^{7}}, -\frac{{\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{362880 \, x^{8}}, \frac{{\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{3628800 \, x^{9}}, -\frac{{\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{39916800 \, x^{10}}, \frac{{\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{479001600 \, x^{11}}, -\frac{{\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{6227020800 \, x^{12}}, \frac{{\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{87178291200 \, x^{13}}, -\frac{{\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{1307674368000 \, x^{14}}, \frac{{\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{20922789888000 \, x^{15}}, -\frac{{\left(x - 16\right)} {\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{355687428096000 \, x^{16}}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, -1, \frac{x - 1}{2 \, x}, -\frac{{\left(x - 2\right)} {\left(x - 1\right)}}{6 \, x^{2}}, \frac{{\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{24 \, x^{3}}, -\frac{{\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{120 \, x^{4}}, \frac{{\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{720 \, x^{5}}, -\frac{{\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{5040 \, x^{6}}, \frac{{\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{40320 \, x^{7}}, -\frac{{\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{362880 \, x^{8}}, \frac{{\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{3628800 \, x^{9}}, -\frac{{\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{39916800 \, x^{10}}, \frac{{\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{479001600 \, x^{11}}, -\frac{{\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{6227020800 \, x^{12}}, \frac{{\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{87178291200 \, x^{13}}, -\frac{{\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{1307674368000 \, x^{14}}, \frac{{\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{20922789888000 \, x^{15}}, -\frac{{\left(x - 16\right)} {\left(x - 15\right)} {\left(x - 14\right)} {\left(x - 13\right)} {\left(x - 12\right)} {\left(x - 11\right)} {\left(x - 10\right)} {\left(x - 9\right)} {\left(x - 8\right)} {\left(x - 7\right)} {\left(x - 6\right)} {\left(x - 5\right)} {\left(x - 4\right)} {\left(x - 3\right)} {\left(x - 2\right)} {\left(x - 1\right)}}{355687428096000 \, x^{16}}\right]
[limit(term,x=oo) for term in terms] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, -1, \frac{1}{2}, -\frac{1}{6}, \frac{1}{24}, -\frac{1}{120}, \frac{1}{720}, -\frac{1}{5040}, \frac{1}{40320}, -\frac{1}{362880}, \frac{1}{3628800}, -\frac{1}{39916800}, \frac{1}{479001600}, -\frac{1}{6227020800}, \frac{1}{87178291200}, -\frac{1}{1307674368000}, \frac{1}{20922789888000}, -\frac{1}{355687428096000}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, -1, \frac{1}{2}, -\frac{1}{6}, \frac{1}{24}, -\frac{1}{120}, \frac{1}{720}, -\frac{1}{5040}, \frac{1}{40320}, -\frac{1}{362880}, \frac{1}{3628800}, -\frac{1}{39916800}, \frac{1}{479001600}, -\frac{1}{6227020800}, \frac{1}{87178291200}, -\frac{1}{1307674368000}, \frac{1}{20922789888000}, -\frac{1}{355687428096000}\right]
sum(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{8178130767479}{22230464256000}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{8178130767479}{22230464256000}
n(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
n(1/e) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442

Hmmm ... do you think this might be equal to \sum_{x = 0}^{\infty} (-1)^x\frac {1}{x!}?
[(-1)^x*1/factorial(x) for x in range(20)] 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, -1, \frac{1}{2}, -\frac{1}{6}, \frac{1}{24}, -\frac{1}{120}, \frac{1}{720}, -\frac{1}{5040}, \frac{1}{40320}, -\frac{1}{362880}, \frac{1}{3628800}, -\frac{1}{39916800}, \frac{1}{479001600}, -\frac{1}{6227020800}, \frac{1}{87178291200}, -\frac{1}{1307674368000}, \frac{1}{20922789888000}, -\frac{1}{355687428096000}, \frac{1}{6402373705728000}, -\frac{1}{121645100408832000}\right]
\newcommand{\Bold}[1]{\mathbf{#1}}\left[1, -1, \frac{1}{2}, -\frac{1}{6}, \frac{1}{24}, -\frac{1}{120}, \frac{1}{720}, -\frac{1}{5040}, \frac{1}{40320}, -\frac{1}{362880}, \frac{1}{3628800}, -\frac{1}{39916800}, \frac{1}{479001600}, -\frac{1}{6227020800}, \frac{1}{87178291200}, -\frac{1}{1307674368000}, \frac{1}{20922789888000}, -\frac{1}{355687428096000}, \frac{1}{6402373705728000}, -\frac{1}{121645100408832000}\right]
sum(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{92079694567171}{250298560512000}
\newcommand{\Bold}[1]{\mathbf{#1}}\frac{92079694567171}{250298560512000}
n(_) 
        
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442
\newcommand{\Bold}[1]{\mathbf{#1}}0.367879441171442