Additional Topics in Analysis, Discrete Mathematics, Geometry, Probability, and Numerical Methods
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For \(n\ge 1\), define \(f_n:[0,1]\to\mathbb{R}\) by \(f_n(x)=x^n\). Which statement is
true
?
Select one option.
A
\(f_n\to 0\) uniformly on \([0,1]\).
B
\(f_n\to 0\) pointwise on \([0,1]\) and the convergence is uniform.
C
\(f_n\) converges pointwise on \([0,1]\) to a limit \(f\) that is not continuous, the convergence is not uniform on \([0,1]\), and \(\int_0^1 f_n(x)\,dx\to 0\).
D
\(f_n\) does not converge pointwise on \([0,1]\).
E
\(\int_0^1 f_n(x)\,dx\to 1\).
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