Need help finding $\lim_{n\to\infty} \lfloor \pi\cdot10^n\rfloor \bmod 10$

Question

I want to evaluate the following limit:

$$\lim_{n\to\infty} \lfloor \pi\cdot10^n\rfloor \bmod 10$$

I think the answer could be something in $\mathbb{Z}_{10}$. Any ideas? :)

Answer 1

The limit doesn't exist.

• The sequence $a_n = \lfloor \pi \cdot 10^n \rfloor \mod 10$ is a sequence of integers.

• A sequence of integers converges if and only if it is eventually constant.

• A number with an eventually constant sequence of digits is rational.

• $\pi$ isn't rational.

Answer 2

\begin{align} \pi & = 3\,.\,1415926535\ldots \\ & \qquad \searrow \\[8pt] 10^5\pi & = 314159\,.\,26535\ldots \\ & ( \text{multiplying by $10^5$ just moves the} \\ & \phantom{(} \text{ decimal point 5 places leftward.)} \\[8pt] \lfloor10^5\pi\rfloor & = 314159 \\[8pt] \lfloor10^5\pi\rfloor & \bmod 10 = 9 \\ & (x\bmod10= \text{the last digit of the positive integer }x.) \\[18pt] \text{So } & \lfloor 10^n\pi\rfloor = \text{the $n$th digit after the decimal point in } \pi. \end{align}