Find $N$, in the decimal expansion of the large number $N=4^{4^{4^4}}$

Question

Find $N$, in the decimal expansion of the large number

$$N=4^{4^{4^4}}$$

Following on from the question I posted yesterday about finding the number of digits

( Find the number of digits, $D$, in the decimal expansion of the large number $N=4^{4^{4^{4}}}$ )

I now wanted to find the $N$ (decimal expansion itself).

could I use this formula possibly?

$$ \sum_{i=1}^\infty 10^{-i} d_i $$

I wanted to work out $N$ because I was faced with the next part of the question

Say a robot could type $10$ billion digits a second! Find the time $T_n$, in years to type out the number $N$ in the previous part of this question.

I don't know how to go about calculating this..

any help is appreciated :)

Answer 1

Given that you know approximately the number of digits $D$, and that there are about $3.16 \cdot 10^7$ seconds in a year, it would take $\frac D{3.16 \cdot 10^{17}}$ years