Strange Pattern in Decimal Expansion I noticed something weird when I was fooling around with my calculator.
I calculated several powers of $30$ of the form $30^{\left(\frac{10^n-1}{10^n}\right)}$ and I noticed a pattern in the fractional part:
$$30^{\left(\frac{9999}{10000}\right)}=29.9897981429$$
$$30^{\left(\frac{99999}{100000}\right)}=29.9989796581$$
$$30^{\left(\frac{999999}{1000000}\right)}=29.9998979643$$
The rest of the powers just keep sticking a $9$ in the tenths place and shifting all the other digits down.
Why is the decimal expansion for $30^{\left(\frac{10^n-1}{10^n}\right)}$ when $n\ge 5$ $29.9\cdots 989796$?
 A: The input is
$$\begin{align}30^{(1-10^{-n})}&=e^{(1-10^{-n})\ln30}=e^{\ln30}e^{-10^{-n}\ln30}\approx30(1-10^{-n}\ln30)\\
&=29.99999999999\dots\\
&\quad -10^{-n}(102.035921\dots)\\
&=29.\underbrace{99\dots99}_{n-3\,9\text{'s}}897964078\dots\end{align}$$
A: Short answer: it's because of the approximation
$$
   (1 + x)^{1/10} \approx 1 + \frac{x}{10}
$$
valid when $x$ is very close to $0$. (This approximation is the linear truncation of the Taylor series $(1 + x)^{1/10} = 1 + \binom{1/10}{1}x + \binom{1/10}{2}x^2 + \binom{1/10}{3}x^3 + \dotsb$.)
A different way of writing this approximation is as
$$
   y^{1/10} \approx \frac{9+y}{10}
$$
valid when $y \approx 1$, by the simple substitution $y = 1+x$.

The operation that takes $30^{\frac{99}{100}}$ to $30^{\frac{999}{1000}}$ to $30^{\frac{9999}{10000}}$ is the map $t \mapsto t^{1/10} \cdot 30^{9/10}$.
When $t$ is very close to $30$, then $\frac{t}{30}$ is very close to $1$, so we have the approximation 
$$
   \left(\frac{t}{30}\right)^{1/10} \approx \frac{9}{10} + \frac{t/30}{10}
$$
which becomes
$$
   t^{1/10} \cdot 30^{9/10} \approx \frac{9}{10} \cdot 30 + \frac{1}{10} \cdot t
$$
when we multiply both sides by $30$. On the right, we have the next term of the sequence of powers: if $t = 30^{99/100}$, then $t^{1/10} \cdot 30^{9/10} = 30^{999/1000}$. On the left, the expression
$$
   \frac{9}{10} \cdot 30 + \frac{1}{10} \cdot t
$$
is a weighted average that says "take a number ten times as close to $30$ as $t$ is". Or to put it another way, the difference $30 - (\frac{9}{10} \cdot 30 + \frac{1}{10} \cdot t)$ simplifies to $\frac1{10}(30 - t)$: a tenth of the distance $t$ was from $30$. When you cut the distance from $30$ in ten, you do this by adding an extra $9$ in the decimal expansion, so the error goes, for example, from
$$
   30 - 30^{0.99999} \approx 0.0010203...
$$
to
$$
   30 - 30^{0.999999} \approx 0.00010203...
$$
