Question

what is the exact value of the function 3pi ?

Answer 1

\(f(x) = 3\pi\)

This is true for all \(x\). So the value of \(f(x)\) is always \(3\pi\)

Answer 2

\[
3\pi = 3\left(4 \sum_{k=0}^\infty \frac{(-1)^k}{2 k+1}\right) = 12 \sum_{k=0}^\infty \frac{(-1)^k}{2 k+1}
\]Since \(\pi\) is a transcendental number, it can only be expressed the sum of an infinity polynomial. There is no finite set of algebraic steps which would result in \(\pi\), as would be the case for an algebraic number.