Limit (6)

• Sep 25th 2009, 03:00 PM
NonCommAlg
Limit (6)
Suppose $f: [0,\infty) \longrightarrow \mathbb{R}$ is continuous and let $a_n=\int_0^1 f(x+n) \ dx, \ \ n \in \mathbb{N}.$ We know that $\lim_{n\to\infty} a_n=L.$ Evaluate $\lim_{n\to\infty} \int_0^1 f(nx) \ dx.$
• Sep 25th 2009, 06:35 PM
PaulRS
First: $\int_0^1 {f\left( {nx} \right)dx} = \tfrac{1}
{n} \cdot \int_0^n {f\left( x \right)dx}$

Now: $
\int_0^n {f\left( x \right)dx} = \sum\limits_{k = 0}^{n - 1} {\int_0^1 {f\left( {x + k} \right)dx} } = \sum\limits_{k = 0}^{n - 1} {a_k }
$
thus we have: $
\int_0^1 {f\left( {nx} \right)dx} = \left(\tfrac{1}
{n} \cdot \sum\limits_{k = 0}^{n - 1} {a_k }\right) \to L
$
- remember Cesàro Mean -