TITLE: Strange Pattern in Decimal Expansion
QUESTION [8 upvotes]: 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 $n\ge 5$ $29.9\cdots 989796$?

REPLY [8 votes]: Consider $$a_n=30^{\left(\frac{10^n-1}{10^n}\right)}\implies \log(a_n)=\frac{10^n-1}{10^n}\log(30)=\left(1-\frac{1}{10^n}\right)\log(30)$$ Then
$$\log(a_{n+1})-\log(a_n)=\log \left(\frac{a_{n+1}}{a_n}\right)=\frac 9{10^{n+1}} \log (30)$$ is probably sufficient o explain the pattern.
You could change $30$ by any number. Trying with $1234.56789$ and using illimited precision, we should have
$$\left(
\begin{array}{cc}
n & a_n \\
 5 & \color{red}{1234}.4800107051986485 \\
 6 & \color{red}{1234.5}591017890121882 \\
 7 & \color{red}{1234.567}0111760860759 \\
 8 & \color{red}{1234.5678}021175804561 \\
 9 & \color{red}{1234.56788}12117577641 \\
 10 & \color{red}{1234.567889}1211757736 \\
 11 &\color{red}{1234.5678899}121175773 \\
 12 & \color{red}{1234.56788999}12117577 \\
 13 & \color{red}{1234.567889999}1211758 \\
 14 & \color{red}{1234.5678899999}121176 \\
 15 & \color{red}{1234.56788999999}12118
\end{array}
\right)$$ Look at the part which is not in red.