Fourier series: Uniform Convergence of Fourier series
Uniform convergence
Consider the following sequence #f_n# #(n=1,2,\ldots)# of real functions defined on the whole real line by
\[ f_n(x) = {{x}\over{n\cdot x^2+1}} \]
The function #f(x) =0 # is the pointwise limit of the sequence. The goal is to determine whether the series converges uniformly to #f#.
Give a simplified expression for the supremum of #\left|f_n(x)-f(x)\right|# for #x\in\mathbb{R}# in terms of #n#.
\[ f_n(x) = {{x}\over{n\cdot x^2+1}} \]
The function #f(x) =0 # is the pointwise limit of the sequence. The goal is to determine whether the series converges uniformly to #f#.
Give a simplified expression for the supremum of #\left|f_n(x)-f(x)\right|# for #x\in\mathbb{R}# in terms of #n#.
| #\sup\{\left|f_n(x)-f(x)\right|\,\mid x\in\mathbb{R}\} = # |
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