Absolute convergence to zero imply convergence to zero?

Question

Given that $\frac{1}{N}\sum_{i=1}^N {|a_i|}$ converges to zero as $N\rightarrow \infty$, does it imply that $\frac{1}{N}\sum_{i=1}^N {a_i}\rightarrow 0$?

I know absolute convergence imply convergence, but in general their limits may not be the same. I wonder if convergence to zero is an exception?

Answer 1

For a given $\epsilon>0$, $$\left|\frac1N \sum_{i=1}^N a_i\right|\le\frac1N \sum_{i=1}^N \left|a_i\right|<\epsilon$$ whenever $N>M(\epsilon)$ for some $M$. Thus, $\frac1N \sum_{i=1}^N a_i$ converges to zero.

Answer 2

Let $\epsilon >0$ be arbitrary ;there exists $m\in \mathbb N$ such that $\forall n\geq m$

$|\dfrac{1}{N}\sum |a_i||<\epsilon$

Now $|\dfrac{1}{N}\sum_{i=1}^Na_i|\leq \dfrac{1}{N}\sum_{i=1}^N|a_i|<\epsilon$ $\forall n\geq m$

Answer 3

This is a basic result from the triangle inequality. i.e. $$|a+b|\le|a|+|b|$$