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
\[ f_n(x) = {{x}\over{2^{n\cdot x^2}}} \]
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#.
\[ f_n(x) = {{x}\over{2^{n\cdot x^2}}} \]
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}\} = # |
Unlock full access
Teacher access
Request a demo account. We will help you get started with our digital learning environment.
Student access
Is your university not a partner?
Get access to our courses via Pass Your Math independent of your university. See pricing and more.
Or visit omptest.org if jou are taking an OMPT exam.
Or visit omptest.org if jou are taking an OMPT exam.
