Chapter 5 Series, Uniform Continuity and More

Theorem 5.1

Suppose \(0 \lt r \lt 1\) and for each positive integer \(n\text{,}\) let \(\displaystyle S_n = \sum_{i=0}^n r^i\text{.}\) Show that the sequence \(S_1, S_2, S_3, \dots\) converges to \(\displaystyle \frac{1}{1-r}\text{.}\)

Definition 5.2

A function \(f\) is uniformly continuous on the set \(M\) if for every \(\epsilon > 0\) there exists a number \(\delta > 0\) so that if \(u,v \in M\) and \(|u-v| \lt \delta\) then \(|f(u)-f(v)| \lt \epsilon\text{.}\)

Theorem 5.3

A function \(f\) is continuous on \([a,b]\) if and only if \(f\) is uniformly continuous on \([a,b]\text{.}\)

This theorem is valid on any closed and bounded (compact) domain.

Problem 5.4

Show that there exists a function \(f\) that is continuous on \((a,b)\) but not uniformly continuous on \((a,b)\text{.}\)

Theorem 5.5

Suppose that \(c\) is a number between \(0\) and \(1\) and \(a_0, a_1, a_2, \dots\) is a sequence of positive numbers and \(a_i \lt c a_{i-1}\) for all \(i=1,2,\dots\) and \(\displaystyle S_n=\sum_{i=0}^n a_i\) for all \(n=1,2,\dots\text{.}\) Show that the sequence \(S_1, S_2, \dots\) converges.

A shorthand for the sequence \(a_1, a_2, a_3, \dots\) is \((a_n)_{n=1}^\infty\text{.}\)

Definition 5.6

If \((a_n)_{n=1}^\infty\) is a sequence then the sequence of partial sums of \((a_n)_{n=1}^\infty\) is the (new) sequence defined by \(\displaystyle S_N =\sum_{n=1}^N a_n, \; N=1,2,3,\dots\text{.}\) If the sequence of partial sums \((S_N)_{N=1}^\infty\) converges then we define the point to which this sequence converges to be the infinite series associated with \((a_n)_{n=1}^\infty\) and denote it by \(\displaystyle \sum_{i=1}^\infty a_i\text{.}\)

Theorem 5.7

If \((a_n)_{n=1}^\infty\) is a sequence and \(\displaystyle \sum_{i=1}^\infty |a_i|\) converges then \(\displaystyle \sum_{i=1}^\infty a_i\) converges.

Theorem 5.8

For each natural number \(n\) define the function \(f_n\) by \(\displaystyle f_n(x) = \sum_{k=0}^n \frac{x^k}{k!}\) for every \(x \in [0,1]\text{.}\) Show that \(f_n\) is continuous on \([0,1]\text{.}\)

Definition 5.9

A function \(f\) is called a Lipschitz function if there exists \(c \geq 0\) such that for every pair \(u,v\) in the domain of \(f,\) \(|f(u)-f(v)| \leq c|u-v|\text{.}\)

Problem 5.10

Show that there exists a function that is Lipschitz on \([a,b]\) but not differentiable on \([a,b]\text{.}\)

Theorem 5.11

Show that every Lipschitz function is uniformly continuous.

Definition 5.12

If \(f_1, f_2, f_3, \dots\) is a sequence of functions with a common domain \(D\) then we say that \(f_1, f_2, f_3,\) converges pointwise on \(D\) if there is a function \(g\) defined on \(D\) so that for each \(x \in D\) the sequence \((f_n(x))_{n=1}^\infty\) converges to \(g(x)\text{.}\)

Definition 5.13

If \(f_1, f_2, f_3, \dots\) is a sequence of functions with a common domain \(D\) then we say that \(f_1, f_2, f_3,\) converges uniformly on \(D\) if there is a function \(g\) defined on \(D\) so that for all \(\epsilon > 0\) there is a natural number \(N\) so that for all natural numbers \(n>N\) and for all \(x \in D\) we have \(|g(x) - f_n(x)| \lt ~\epsilon\text{.}\)

Problem 5.14

Show there is a sequence of continuous functions, \(f_1, f_2, f_3, \dots\) converging pointwise to a function that is not continuous.

Problem 5.15

Show there is a sequence of differentiable functions, \(f_1, f_2, f_3, \dots\) converging pointwise to a function that is not continuous.

Theorem 5.16

Show that if \(f_1, f_2, f_3, \dots\) is a sequence of continuous functions converging uniformly to the function \(g\) then \(g\) is continuous.