A calculus problem by Star Fall
Calculus
Level
5
How many of the following statement(s) is/are wrong ?
- The pointwise limit of a sequence of continuous functions is continuous.
- Every continuous function defined on closed real intervals is Riemann integrable .
- Real power series are uniformly convergent within compact subintervals of their interval of convergence .
- Every real-valued smooth (infinitely differentiable ) function admits a Maclaurin series expansion which converges to the function on some open subset of its domain.
- A convergent series whose terms are differentiable functions of x can always be differentiated termwise to obtain the derivative of the limit function.
3
2
1
4
0
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1 solution
The incorrect statements are #1, #4 and #5.
A counterexample to 1 is the sequence of functions f n ( x ) = x n on the interval [ 0 , 1 ] . The pointwise limit, as is easily checked, is not continuous.
4 is correct for complex-valued smooth (holomorphic) functions, but it is false for real-valued smooth functions: consider the function f ( x ) = e − 1 / x 2 . It is easy to check that every derivative of f ( x ) vanishes at x = 0 , but f is not identically zero, whereas its Taylor expansion turns out to be.
5 requires uniform convergence of the series, in general, it is true for power series within their interval of convergence. However, it is not generally true; consider the sum of sin ( k x ) / k as k ranges over the positive integers. This sum is convergent to a differentiable function (this is a well known fact), but termwise differentiation gives something which does not even converge.