5.9.E: Problems on Convergence in Differentiation and Integration
Complete all proof details in Theorems 1 and \(3,\) Corollaries 1 and \(2,\) and Note \(3 .\)
Show that assumptions (a) and (c) in Theorem 1 can be replaced by \(F_{n} \rightarrow F\) (pointwise) on \(I\). (In this form, the theorem applies to incomplete spaces \(E\) as well.)
\(\left.\text { [Hint: } F_{n} \rightarrow F(\text { pointwise } e), \text { together with formula ( } 3\right),\) implies \(F_{n} \rightarrow F\) (uniformly) \(\text { on } I .]\)
Show that Theorem 1 fails without assumption \((\mathrm{b}),\) even if \(F_{n} \rightarrow F\) (uniformly) and if \(F\) is differentiable on \(I .\)
[Hint: For a counterexample, try \(F_{n}(x)=\frac{1}{n} \sin n x,\) on any nondegenerate \(I .\) Verify that \(F_{n} \rightarrow 0\) (uniformly), yet (b) and assertion (iii) fail.
Prove Abel's theorem (Chapter \(4, §13,\) Problem 15) for series
\[
\sum a_{n}(x-p)^{n} ,
\]
with all \(a_{n}\) in \(E^{m}\left(^{*} \text { or in } C^{m}\right)\) but with \(x, p \in E^{1}\).
[Hint: Split \(a_{n}(x-p)^{n}\) into components.]
Prove Corollary 3.
\(\text { [Hint: By Abel's theorem (see Problem } 4),\) we may put
\[
\sum_{n=0}^{\infty} a_{n}(x-p)^{n}=F(x)
\]
\(\left.\text { uniformly on }\left[p, x_{0}\right] \text { (respectively, }\left[x_{0}, p\right]\right) .\) This implies that \(F\) is relatively continuous at \(x_{0} .\) (Why?) So is \(f,\) by assumption. Also \(f=F\) on \(\left[p, x_{0}\right)\left(\left(x_{0}, p\right]\right) .\) Show that
\[
f\left(x_{0}\right)=\lim f(x)=\lim F(x)=F\left(x_{0}\right)
\]
as \(x \rightarrow x_{0}\) from the left (right).]
In the following cases, find the Taylor series of \(F\) about 0 by integrating the series of \(F^{\prime} .\) Use Theorem 3 and Corollary 3 to find the convergence radius \(r\) and to investigate convergence at \(-r\) and \(r .\) Use \((\mathrm{b})\) to find a formula for \(\pi .\)
(a) \(F(x)=\ln (1+x)\);
(b) \(F(x)=\arctan x\);
(c) \(F(x)=\arcsin x\).
Prove that
\[
\int_{0}^{x} \frac{\ln (1-t)}{t} d t=\sum_{n=1}^{\infty} \frac{x^{n}}{n^{2}} \quad \text { for } x \in[-1,1] .
\]
[Hint: Use Theorem 3 and Corollary \(3 .\) Take derivatives of both sides.]