Math problem where graphing calculator may slow you down...

[Thomas Klemm]

(08-17-2014 10:25 PM)Bunuel66 Wrote:  Nothing more than a pen, a 'small' paper and the first scientific calculator you can grasp around you

Here's how you can figure out with a 4-banger whether \(e^{\frac{5}{4}\pi} > 50\):

AC
3.141592654
÷ 1024
+ 1 =
× =
× =
× =
× =
× =
× =
× =
× =
M+
× =
× =
× MR =

The result is:
50.44976377

As \((1+\frac{x}{n})^n \leq e^x\), we can conclude that indeed:
\[
\begin{align}
50 & < 50.44976377 \doteq (1+\frac{\pi}{1024})^{1024+256} \\
& = (1+\frac{\pi}{1024})^{1024 \times 1.25} \\
& = ((1+\frac{\pi}{1024})^{1024})^{1.25} \\
& \leq (e^\pi)^{1.25} \\
& = e^{\pi \times 1.25} \\
& = e^{\frac{5}{4}\pi}
\end{align}
\]

Cheers
Thomas