How can we get Nth digit of any irrational number in base 2 using Taylor expansion?

Question

How can we get Nth digit of any irrational number in base   2 using Taylor expansion?

Elliana Shelton · Accepted Answer

There is not a general method to compute any irrational number, in fact there is provably no method to compute "most" irrational numbers (most in terms of measure). Also, for a lot of constants there is no known method to directly convert its Taylor series to the terms of its binary or decimal expansion. For most numbers, the only approach is likely to compute the number to a desired accuracy and then round its decimals.
For example, with

π = 4 \arctan 1

, we can express it with the Taylor series,

\arctan x = \sum_{i = 0}^{\infty} \frac{{(- 1)}^{i} x^{2 i + 1}}{2 i + 1}

, but so far, there is no known way to convert this to a "spigot algorithm".

How can we get Nth digit of any irrational number

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