Approximation of $\cos{1}$ , (1 radiant)

Question

I have found an approximation of e to the first $5$ million digits here. Is there an analogous approximation (or to as many digits as possible) of $\cos{1}$ ($1$ radian)?

Thanks in advance :)

Answer 1

A computer algebra system (Maple, Mathematica etc.) should be able to compute $5$ million digits easily.

Answer 2

If you have a platform that can do computations with 5-million-digit accuracy then you can compute the first 5 million digits of $\cos(1)$ using the power series $$\cos(t)=\sum_{j=0}^\infty \frac{(-1)^jt^{2j}}{(2j)!}.$$

To figure out how many terms you need, note that if you're computing $\cos(1)$ then the tail, that is the error after $N-1$ terms, is smaller than $$E=\sum_{j=N}^\infty\frac1{(2j)!}<\sum_{j=2N}^\infty\frac1{j!}.$$ If you note that $(j+1)!>2(j!)$ for $j>1$ it follows that $$E\le\frac2{(2N)!},$$so you only need $N$ terms, if $N$ is large enough that $$\frac2{(2N)!}<10^{-5,000,000}.$$