series with irrational numbers

Question

Yeah the title says everything I will explain this quick if someone is so smart and nice than he has my ammiration! Here you are :: if we take an irrational number like π or e or whatever and we write this π+π-π… (ecc) at infinity of this series what could possibly come out?? I hope somebody can explain this thanks in advance to everybody!!

Answer 1

This series is analagous to $1-1+1-1+ \cdots = \sum_{n=0}^{\infty} (-1)^n$. This is not a convergent series, because the sequence of martial sums does not converge, it just keeps alternating between $0$ and $1$.

$S_1 = 1$

$S_2 = 1-1=0$

$S_3 = 1-1+1=1$

$\vdots$

EDIT: Thanks to Clement C.

If you allow yourself weaker notions of convergence (which you usually study in upper division classes) you can assign a value to some divergent series.

Wikipedia Article

Answer 2

So what you are looking for is a series $$\sum_{n=1}^{\infty} (-1)^{a_n} x$$

where $a_n$ is a sequence of natural numbers and $x$ is an irrational number.

This series always fails the limit test, so it is divergent