Weighted uniform convergence of Taylor series of exponential function

Solution 1:

Credits should go to Landscape.

Define $$r_n(x)=\sum_{k=n+1}^\infty (-1)^k\frac{x^k}{k!}$$

Note that by Taylor's theorem with Lagrange's form of the remainder we can write $$r_n(x)=(-1)^{n+1}e^{-x'}\frac{x^{n+1}}{(n+1)!}$$

where $x'$ is positive. It follows $$e^{-x}|r_n(x)|\leq e^{-x}\frac{x^{n+1}}{(n+1)!}$$

Easy verification shows the last function has absolute maximum at $x=n+1$. But $$\frac{1}{{(n + 1)!}}{\left( {\frac{{n + 1}}{e}} \right)^{n + 1}} \sim \frac{1}{{\sqrt {2\pi \left( {n + 1} \right)} }}$$ by Stirling, so convergence is indeed uniform. $\quad \Box$

Solution 2:

Thanks, this was a fun problem.

From the integral representation

$$ \sum_{k=0}^{n} \frac{x^k}{k!} = \frac{1}{n!} \int_0^\infty (x+t)^n e^{-t} \,dt \tag1 $$

we can derive the expression

$$ e^{-x} \sum_{k=0}^{n} \frac{(-x)^k}{k!} = e^{-2x} - \frac{e^{-2x} (-x)^{n+1}}{n!} \int_0^1 t^n e^{xt}\,dt. \tag2 $$

Now

$$ \int_0^1 t^n e^{xt}\,dt \leq e^x \int_0^1 t^n\,dt = \frac{e^x}{n+1}, \tag3 $$

so that

$$ \begin{align} \left|\frac{e^{-2x} (-x)^{n+1}}{n!} \int_0^1 t^n e^{xt}\,dt\right| &\leq \frac{e^{-x} x^{n+1}}{(n+1)!} \\ &\leq \frac{e^{-n-1} (n+1)^{n+1}}{(n+1)!} \\ &\sim \frac{1}{\sqrt{2\pi n}} \end{align} \tag4 $$

by Stirling's formula.

Added by 40 votes for those interested in the derivation of (2) from (1): $$ \sum_{k=0}^{n} \frac{(-x)^k}{k!} = \frac{1}{n!} \int_0^x (t-x)^n e^{-t} \,dt + \frac{1}{n!} \int_x^\infty (t-x)^n e^{-t} \,dt \tag{A} $$ Substitute $u=t-x$ in the second integral on the right of (A): $$\frac{1}{n!}\int_x^\infty (t-x)^n e^{-t} \,dt =\frac{1}{n!}\int_0^\infty u^n e^{-u-x} \,dt = e^{-x} \tag{B}$$ Substitute $u=1-t/x$ in the first integral on the right of (A), noting that $(t-x)^n=(-x)^n u^n$ and $dt=(-x)du$: $$\frac{1}{n!} \int_0^x (-x+t)^n e^{-t} \,dt = \frac{(-x)^{n+1}}{n!} \int_0^1 u^n e^{xu-x} \,du = \frac{e^{-x}(-x)^{n+1}}{n!} \int_0^1 u^n e^{xu} \,du \tag{C} $$ Adding (B) and (C), identity (2) follows.

Solution 3:

I'd like to provide another solution which is a mixture of Antonio's and Landscape's. One can also write $$\left|r_n(x)\right|=\int_0^x {e^{-t}}\frac{(x-t)^n}{n!}dt$$

by virtue of Taylor's theorem with the integral remainder. But then again $$\left| {{r_n}(x)} \right| \leqslant \int_0^x {\frac{{{{(x - t)}^n}}}{{n!}}dt} = \int_0^x {\frac{{{u^n}}}{{n!}}du} = \frac{x^{n + 1}}{{\left( {n + 1} \right)!}}$$

and Strling does the job.