Geometry

Topology

Deep Dive: Pointwise and Uniform Convergence 1

         

A sequence of functions fn(x)f_n(x) is drawn in stages. Each stage in the sequence consists of a connected path of semicircles:

f1(x)f_1(x) consists of 1 semicircle of radius 1,

f2(x)f_2(x)consists of 2 semicircles of radius 12,\frac{1}{2},

f3(x)f_3(x)consists of 4 semicircles of radius 14,\frac{1}{4},

and so on.

As nn increases and this process continues indefinitely, what happens to the lengths of fn(x)f_n(x) over the interval [0,2]?[0,2]?

True or False?

Let fn(x)f_n(x) be the function on [0,2][0,2] whose graph is the Stage nn semicircles. Then fn(13)0,f_n\left(\frac{1}{3}\right) \rightarrow 0, i.e.,

limnfn(13)=0.\lim_{n \to \infty} f_n\left(\frac{1}{3}\right) = 0.

True or False?

Let fn(x)f_n(x) be the function on [0,2][0,2] whose graph is the Stage nn semicircles. Then for all xx in [0,2],[0,2], fn(x)0,f_n\left(x\right) \rightarrow 0, i.e.,

limnfn(x)=0.\lim_{n \to \infty} f_n\left(x\right) = 0.

Note. Another way of phrasing the question is: "Does the sequence {fn(x)}\{f_n(x)\} converge pointwise to f(x)=0 f(x) = 0 on the interval [0,2]?[0,2]?

True or False?

Let fn(x)f_n(x) be the function on [0,2][0,2] whose graph is the Stage nn semicircles. Then for all xx in [0,2][0,2] where they are defined, the derivatives (fn)(x)0,(f_n)'\left(x\right) \rightarrow 0, i.e.,

limn(fn)(x)=0.\lim_{n \to \infty} (f_n)'\left(x\right) = 0.

Note. 0 is the derivative of the function f(x)=0f(x) = 0 that the functions fnf_n approach.

True or False?

Suppose {fn}\{f_n\} and ff are functions on an interval I,I, and L(g)L(g) represents the length of the curve gg on I.I.

If fn(x)f(x)f_n(x) \to f(x) for all xI,x \in I, then L(fn)L(f).L(f_n) \to L(f).

Alternate phrasings:

If limnfn(x)=f(x)\lim_{n \to \infty} f_n(x) = f(x) for all xI,x \in I, then limnL(fn)=L(f).\lim_{n \to \infty}L(f_n) = L(f).

If the sequence of curves approach ff at each point, then the lengths of the curves approach the length of f.f.

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