Series

Abstract

We study infinite sums (series) of numbers and of functions. Among the topics are products of series, the Riemann rearrangement theorem, and the theory of power series. As applications we construct a space filling curve, construct a continuous nowhere differentiable functions, and prove the Weierstrass Approximation Theorem.

Appendices

Problems

1.1 Problems for Sect. 10.1

• 1. Suppose $\sum a_{n}$ is convergent. Let $b_{n}:=a_{2n-1}+a_{2n}$ for all $n.$ Show $\sum b_{n}$ is convergent.

• 2. Suppose $a_{n}\geq 0,$ $b_{n}\geq 0,$ $\sum a_{n}^{2}$ is convergent, and $\sum b_{n}^{2}$ is convergent. Show $\sum a_{n}b_{n}$ is convergent. [Hint: $\left (a-b\right )^{2}\geq 0.$]

• 3. Look up Raabe’s test and give a proof of this test.

• 4. If $a_{n}=b_{n}:=(-1)^{n+1}1/\sqrt {n+1},$ then $\sum a_{n}$ and $\sum b_{n}$ are convergent by the Alternating Series Test. Let $c_{n}$ be determined by (10.2). Show that $\sum c_{n}$ is divergent.

• 5. If $\sum a_{n}$ is absolutely convergent and $\sum b_{n}$ is convergent. Let $c_{n}$ be determined by (10.2). Must $\sum c_{n}$ be convergent?

• 6. Let $a_{k}:=\frac {1}{k}-\log \left (1+\frac {1}{k}\right ).$

  • a. Show $a_{k}>0.$

  • b. Prove $\sum _{1}^{\infty }a_{k}$ is convergent. [The sum $\gamma :=\sum _{k=1}^{\infty }a_{k}$ is Euler’s constant.]

• 7. Suppose $a_{1}\geq a_{2}\geq a_{3}\geq \cdots \geq 0$ and let $\left (b_{k}\right )$ be a sequence of complex numbers such that the sequence of partial sums $B_{n}:=\sum _{k=1}^{n}b_{k}$ is convergent. Show $\sum _{k=1}^{\infty }a_{k}b_{k}$ is convergent. [Hint: Modify the proof of Dirichlet’s Test.]

• 8 Suppose $a_{n}\geq 0$ and $\sum a_{n}$ is convergent.

  • a. Assuming there is a real number $B$ such that $\left |b_{n}\right |\leq B$ for all $n,$ show $\sum a_{n}b_{n}$ is convergent.

  • b. Assuming $\sum b_{n}$ is convergent, show $\sum a_{n}b_{n}$ is convergent.

1.2 Problems for Sect. 10.2.

• 1. Let $f_{n}$ be a sequence of integrable functions on the interval $[a,b].$ If $\sum _{n=1}^{\infty }f_{n}$ converges uniformly to $f$ on $[a,b],$ then $f$ is integrable on $[a,b]$ and

  $$ \int_{a}^{b}\sum_{n}f_{n}=\sum_{n}\int_{a}^{b}f_{n}. $$

• 2. Show $\sum _{k=0}^{\infty }x^{k}\sin \left (kx\right )$ is convergent for $x\in ]-1,1[.$

1.3 Problems for Sect. 10.3

• 1. Show that $y$ is not differentiable at any point.

1.4 Problems for Sect. 10.4

• 1. Show the Cauchy–Hadamard formula implies the root test.

• 2. Use Lemma 10.4.4 and that a uniform limit of continuous functions is a continuous function to show: If $\sum _{n=0}^{\infty }a_{n}x^{n}$ has radius of convergence $R,$ then $f(x)=\sum _{n=0}^{\infty }a_{n}x^{n}$ is continuous on the open disk $\{x\mid |x|<R\}.$

• 3. Suppose $\sum _{n=0}^{\infty }a_{n}x^{n}=\sum _{n=0}^{\infty }b_{n}x^{n}$ on some interval $]-\delta ,\delta [.$ Show that $a_{n}=b_{n}$ for all $n.$

• 4. Find the radius of convergence of $\sum _{k=0}^{\infty }k^{k}z^{k}.$

• 5. Find the radius of convergence of $\sum _{k=0}^{\infty }\frac {1}{k!}z^{k}.$

• 6. Find the radius of convergence of $\sum _{k=0}^{\infty }k!z^{k}.$

1.5 Problems for Sect. 10.5

• 1. We did not prove a change of variables formula for improper integrals. Carefully verify the claim: $\int _{-\infty }^{\infty }nE(nx)\, dx=\int _{-\infty }^{\infty }E(u)\, du$ in the proof of Lemma 10.5.6.

Solutions and Hints for the Exercises

Exercise 10.1.1. $\sum _{k=1}^{n}\left (ax_{k}+by_{k}\right )=a\sum _{k=1}^{n}x_{k}+b\sum _{k=1}^{n}y_{k}$ the right hand side converges to $a\sum _{k=1}^{\infty }x_{k}+b\sum _{k=1}^{\infty }y_{k}.$

Exercise 10.1.5. (1) $x_{n+1}=s_{n+1}-s_{n}\to s-s=0.$ (2) $\sum _{k=N+1}^{\infty }x_{k}=\sum _{k=1}^{\infty }x_{k}-s_{N}\to \sum _{k=1}^{\infty }x_{k}-\sum _{k=1}^{\infty }x_{k}=0.$ (3) $s_{n+1}=s_{n}+x_{n+1}$ so $\left (s_{n}\right )$ is increasing, hence convergent iff bounded.

Exercise 10.1.7. Since $\left |\sum _{k=n}^{m}x_{k}\right |\leq \sum _{k=n}^{m}\left |x_{k}\right |,$ this follows from the Cauchy criterion.

Exercise 10.1.9. We may assume the inequality holds for all $n.$ It follows that $|x_{n+1}|\leq |x_{1}|r^{n},$ hence an application of the Dominated Convergence Theorem completes the proof.

Exercise 10.1.10. Similar to the previous exercise.

Exercise 10.1.12. This really is similar to Example 10.2. The change is we want the inequality to go the other way, so the sequence of partial sums will be unbounded. This can be accomplished by using $\frac {1}{n^{a}}\geq \frac {1}{2^{a(k+1)}}$ for $n$ between $2^{k}$ and $2^{k+1}-1.$

Exercise 10.1.14. Suppose $\int _{1}^{\infty }f$ is convergent. Since $f$ is decreasing $s=\sum _{k=1}^{n}$ $f(k)\mathbb {1}_{]k,k+1[}$ is a lower step function for $f$ on the interval $[1,n+1].$ Hence $\sum _{k=1}^{n}x_{k}=\sum s\leq \int _{1}^{n+1}f.$ Consequently, the partial sums $\left (\sum _{k=1}^{n}x_{k}\right )$ is bounded by $\int _{1}^{\infty }f.$ The converse is similar, using $S=\sum _{k=2}^{n}f(k)\mathbb {1}_{]k-1,k[}$ is an upper step function for $f$ on the interval $[1,n].$

Exercise 10.1.17. Show $-a_{n+1}B_{n}+\alpha _{n+1}B_{n+1}=a_{n+1}b_{n+1}-a_{n+2}B_{n+1}.$

Exercise 10.3.3. Similar to what we did for $x.$

Exercise 10.3.3 For each $n$ either $a_{n}<t$ or $a_{n}=t$. For $n$ with $a_{n}\neq t$ we have

$$ \frac{f\left(b_{n}\right)-f\left(a_{n}\right)}{b_{n}-a_{n}}=\frac{b_{n}-t}{b_{n}-a_{n}}\frac{f\left(b_{n}\right)-f\left(t\right)}{b_{n}-t}+\frac{t-a_{n}}{b_{n}-a_{n}}\frac{f\left(t\right)-f\left(a_{n}\right)}{t-a_{n}}. $$

Exercise 10.4.7. This is a consequence of Theorem 9.2.6.

Exercise 10.4.8. $(1+\varepsilon )^{n}\geq n\varepsilon$ [why?]. So $n^{1/n}\leq (1/\varepsilon )^{1/n}(1+\varepsilon ).$ Now use that $(1/\varepsilon )^{1/n}\to 1$ as $n\to \infty .$ Other proofs are also possible.

Exercise 10.4.9. Follows from the previous exercise and the Cauchy–Hadamard formula.

Exercise 10.4.10. This is a simple consequence of Corollary 9.2.9.