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Math Help - Limit (6)

  1. #1
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    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.
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  2. #2
    Super Member PaulRS's Avatar
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    First: \int_0^1 {f\left( {nx} \right)dx}  = \tfrac{1}<br />
{n} \cdot \int_0^n {f\left( x \right)dx}

    Now: <br />
\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 } <br />
thus we have: <br />
\int_0^1 {f\left( {nx} \right)dx}  = \left(\tfrac{1}<br />
{n} \cdot \sum\limits_{k = 0}^{n - 1} {a_k }\right)  \to L<br />
- remember CesÓro Mean -
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