Limit (6)

**NonCommAlg** · September 25th 2009, 03:00 PM

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 $\lim_{n\to\infty} \int_0^1 f(nx) \ dx.$

**PaulRS** · September 25th 2009, 06:35 PM

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 -

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