Numerical Sequences and Series

Prove that convergence of $\left\{ s_{n}\right\}$ implies convergence of $\left\{ \left|s_{n}\right|\right\}$. Is the converse true?

Calculate $\lim \left(\sqrt{n^{2}+n}-n\right)$.

If $s_{1} = \sqrt{2}$, and

prove that $\left\{ s_{n}\right\}$ converges, and that $s_{n} < 2$ for $n = 1,2,3, \ldots$.

Find the upper and lower limits of the sequence $\left\{ s_{n}\right\}$ defined by

For any two real sequences $\left\{ a_{n}\right\} ,\left\{ b_{n}\right\}$, prove that

provided the sum on the right is not of the form $\infty -\infty$.

Investigate the behavior (convergence or divergence) of $\Sigma a_{n}$ if

Prove that the convergence of $\Sigma a_{n}$ implies the convergence of

if $a_{n} \geq 0$.

If $\Sigma a_{n}$ converges, and if $\left\{ b_{n}\right\}$ is monotonic and bounded, prove that $\Sigma a_{n} b_{n}$ converges.

Find the radius of convergence of each of the following power series:

Suppose that the coefficients of the power series $\sum a_{n} z^{n}$ are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1.

Suppose $a_{n} > 0, s_{n} = a_{1}+\cdots +a_{n}$, and $\Sigma a_{n}$ diverges.

Suppose $a_{n} > 0$ and $\Sigma a_{n}$ converges. Put

Prove that the Cauchy product of two absolutely convergent series converges absolutely.

If $\left\{ s_{n}\right\}$ is a complex sequence, define its arithmetic means $\sigma_{n}$ by

Definition 3.21 can be extended to the case in which the $a_{n}$ lie in some fixed $\mathbb {R}^{k}$. Absolute convergence is defined as convergence of $\Sigma \left|\mathbf{a}_{n}\right|$. Show that Theorems $3.22,3.23,3.25(a), 3.33,3.34,3.42,3.45,3.47$, and 3.55 are true in this more general setting. (Only slight modifications are required in any of the proofs).

Definition 3.21: given a sequence $\left\{ a_{n}\right\}$, with $\left\{ a_{n}\right\}$ one associates the sequence of partial sums $s_{n} = \sum_{k=1}^{n} a_{k}$; the series $\sum_{n=1}^{\infty } a_{n}$ (also written $\Sigma a_{n}$) converges to $s$, written $\sum_{n=1}^{\infty } a_{n} = s$, if $\left\{ s_{n}\right\}$ converges to $s$; if $\left\{ s_{n}\right\}$ diverges, the series is said to diverge.

The cited results, as stated in the book for (real or complex) numbers.

Theorem 3.22: $\Sigma a_{n}$ converges if and only if for every $\varepsilon > 0$ there is an integer $N$ such that $\left|\sum_{k=n}^{m} a_{k}\right| \leq \varepsilon$ if $m \geq n \geq N$.

Theorem 3.23: If $\Sigma a_{n}$ converges, then $\lim_{n \rightarrow \infty } a_{n} = 0$.

Theorem 3.25(a): If $\left|a_{n}\right| \leq c_{n}$ for $n \geq N_{0}$, where $N_{0}$ is some fixed integer, and if $\Sigma c_{n}$ converges, then $\Sigma a_{n}$ converges.

Theorem 3.33 (Root Test): Given $\Sigma a_{n}$, put $\alpha = \limsup_{n \rightarrow \infty } \sqrt[n]{\left|a_{n}\right|}$. Then (a) if $\alpha < 1$, $\Sigma a_{n}$ converges; (b) if $\alpha > 1$, $\Sigma a_{n}$ diverges; (c) if $\alpha = 1$, the test gives no information.

Theorem 3.34 (Ratio Test): The series $\Sigma a_{n}$ (a) converges if $\limsup_{n \rightarrow \infty } \left|a_{n+1} / a_{n}\right| < 1$, (b) diverges if $\left|a_{n+1} / a_{n}\right| \geq 1$ for all $n \geq n_{0}$, where $n_{0}$ is some fixed integer.

Theorem 3.42: Suppose (a) the partial sums $A_{n}$ of $\Sigma a_{n}$ form a bounded sequence; (b) $b_{0} \geq b_{1} \geq b_{2} \geq \cdots$; (c) $\lim_{n \rightarrow \infty } b_{n} = 0$. Then $\Sigma a_{n} b_{n}$ converges.

Theorem 3.45: If $\Sigma a_{n}$ converges absolutely, then $\Sigma a_{n}$ converges.

Theorem 3.47: If $\Sigma a_{n} = A$, and $\Sigma b_{n} = B$, then $\Sigma \left(a_{n}+b_{n}\right) = A+B$, and $\Sigma c a_{n} = c A$, for any fixed $c$.

Theorem 3.55: If $\Sigma a_{n}$ is a series of complex numbers which converges absolutely, then every rearrangement of $\Sigma a_{n}$ converges, and they all converge to the same sum. (A rearrangement is $\Sigma a_{n}^{\prime }$ with $a_{n}^{\prime } = a_{k_{n}}$, where $\left\{ k_{n}\right\}$ is a sequence in which every positive integer appears once and only once.)

Fix a positive number $\alpha$. Choose $x_{1} > \sqrt{\alpha }$, and define $x_{2}, x_{3}, x_{4}, \ldots$, by the recursion formula

Fix $\alpha > 1$. Take $x_{1} > \sqrt{ } \bar{\alpha }$, and define

Replace the recursion formula of Exercise 16 by

where $p$ is a fixed positive integer, and describe the behavior of the resulting sequences $\left\{ x_{n}\right\}$.

Associate to each sequence $a = \left\{ \alpha_{n}\right\}$, in which $\alpha_{n}$ is 0 or 2, the real number

Prove that the set of all $x(a)$ is precisely the Cantor set described in Sec. 2.44.

Suppose $\left\{ p_{n}\right\}$ is a Cauchy sequence in a metric space $X$, and some subsequence $\left\{ p_{n t}\right\}$ converges to a point $p \in X$. Prove that the full sequence $\left\{ p_{n}\right\}$ converges to $p$.

Prove the following analogue of Theorem $3.10(b)$ : If $\left\{ E_{n}\right\}$ is a sequence of closed nonempty and bounded sets in a complete metric space $X$, if $E_{n} \supset E_{n+1}$, and if

then $\bigcap_{1}^{\infty } E_{n}$ consists of exactly one point.

Suppose $X$ is a nonempty complete metric space, and $\left\{ G_{n}\right\}$ is a sequence of dense open subsets of $X$. Prove Baire's theorem, namely, that $\bigcap_{1}^{\infty } G_{n}$ is not empty. (In fact, it is dense in $X$). Hint: Find a shrinking sequence of neighborhoods $E_{n}$ such that $E_{n} \subset G_{n}$, and apply Exercise 21.

Suppose $\left\{ p_{n}\right\}$ and $\left\{ q_{n}\right\}$ are Cauchy sequences in a metric space $X$. Show that the sequence $\left\{ d\left(p_{n}, q_{n}\right)\right\}$ converges. Hint: For any $m, n$,

it follows that

is small if $m$ and $n$ are large.

Let $X$ be a metric space.

Let $X$ be the metric space whose points are the rational numbers, with the metric $d(x, y) = \left|x-y\right|$. What is the completion of this space? (Compare Exercise 24).