91Ó°ÊÓ

Examine the uniform limiting behavior of the sequence of functions $$ f_{n}(x)=\frac{x^{n}}{1+x^{n}} . $$ On what sets can you determine uniform convergence?

Prove that $$ \lim _{n \rightarrow \infty} \int_{\frac{\pi}{2}}^{\pi} \frac{\sin n x}{n x} d x=0 $$

Show that the function \(f(x)=(x-a)^{\frac{1}{3}}\) has an infinite derivative at \(x=a\) and a finite derivative elsewhere.

Can a sequence of discontinuous functions converge uniformly on an interval to a continuous function?

Can the sequence of functions \(f_{n}(x)=\frac{\sin n x}{n^{3}}\) be differentiated term by term? How about the series \(\sum_{k=1}^{\infty} \frac{\sin k x}{k^{3}} ?\)

Let \(f_{n}\) be a sequence of functions converging pointwise to a function \(f\) on the interval \([0,1]\). Suppose that each function \(f_{n}\) is convex on \([0,1]\). Show that the convergence is uniform on any interval \([a, b] \subset(0,1)\). Need it be uniform on \([0,1] ?\)

Give an example of a function \(F\) that is differentiable on \(\mathbb{R}\) such that the sequence $$ f_{n}(x)=n(F(x+1 / n)-F(x)) $$ converges pointwise but not uniformly to \(F^{\prime}\).

Examine the pointwise limiting behavior of the sequence of functions $$ f_{n}(x)=\frac{x^{n}}{1+x^{n}} . $$

Let \(q_{1}, q_{2}, q_{3}, \ldots\) be an enumeration of \(\mathbb{Q} \cap[0,1] .\) For each \(k \in \mathbb{N}\) let \(f_{k}(x)=\left(x-q_{k}\right)^{\frac{1}{3}} .\) Let $$ f(x)=\sum_{k=1}^{\infty} \frac{f_{k}(x)}{10^{k}} $$ Show that the series defining \(f\) converges uniformly.

Show that the logarithm function can be expressed as the pointwise limit of a sequence of "simpler" functions, $$ \ln x=\lim _{n \rightarrow \infty} n(\sqrt[n]{x}-1) $$ for every point in its domain. If the answer to our three questions for this particular limit is affirmative, what can you say about the continuity of the logarithm function? What would be its derivative? What would be \(\int_{1}^{2} \ln x d x ?\)