Chapter 2

Suppose \(\left\\{x_{n}\right\\}\) is such that \(\liminf x_{n}=-\infty,\) limsup \(x_{n}=\infty\). a) Show that \(\left\\{x_{n}\right\\}\) is not convergent, and also that neither \(\lim x_{n}=\infty\) nor \(\lim x_{n}=-\infty\) is true. b) Find an example of such a sequence.

Let s_ be the kth partial sum of \(\sum x_{n}\). a) Suppose that there exists an \(m \in \mathbb{N}\) such that \(\lim _{k \rightarrow \infty} s_{m k}\) exists and \(\lim x_{n}=0 .\) Show that \(\sum x_{n}\) converges. b) Find an example where \(\lim _{k \rightarrow \infty} s_{2 k}\) exists and \(\lim x_{n} \neq 0\) (and therefore \(\sum x_{n}\) diverges). c) (Challenging) Find an example where \(\lim x_{n}=0,\) and there exists a subsequence \(\left\\{s_{k}\right\\}\) such that \(\lim _{j \rightarrow \infty} s_{k}\) exists, but \(\sum x_{n}\) still diverges.

Let \(\left\\{x_{n}\right\\}\) be a convergent monotone sequence. Suppose there exists a \(k \in \mathbb{N}\) such that $$\lim _{n \rightarrow \infty} x_{n}=x_{k}$$ Show that \(x_{n}=x_{k}\) for all \(n \geq k\).

Suppose \(x_{1}:=c\) and \(x_{n+1}:=x_{n}^{2}+x_{n} .\) Show that \(\left\\{x_{n}\right\\}\) converges if and only if \(-1 \leq c \leq 0,\) in which case it converges to 0 .

Suppose \(\left\\{c_{n}\right\\}\) is any sequence. Prove that for any \(r \in(0,1)\) there exists a strictly increasing sequence \(\left\\{n_{k}\right\\}\) of natural numbers \(\left(n_{k+1}>n_{k}\right)\) such that $$ \sum_{k=1}^{\infty} c_{k} x^{n_{k}} $$ converges absolutely for all \(x \in[-r, r] .\)

Let \(\left\\{x_{n}\right\\}\) be a sequence. a) Show that \(\lim x_{n}=\infty\) if and only if \(\liminf x_{n}=\infty\). b) Then show that \(\lim x_{n}=-\infty\) if and only if limsup \(x_{n}=-\infty\). c) If \(\left\\{x_{n}\right\\}\) is monotone increasing, show that either \(\lim x_{n}\) exists and is finite or \(\lim x_{n}=\infty .\) In either case, \(\lim x_{n}=\sup \left\\{x_{n}: n \in \mathbb{N}\right\\}\)

Find a convergent subsequence of the sequence \(\left\\{(-1)^{n}\right\\}\).

Prove \(\lim _{n \rightarrow \infty}\left(n^{2}+1\right)^{1 / n}=1\)

Suppose \(\left\\{x_{n}\right\\}\) is a decreasing sequence of positive numbers. The proof of convergence/divergence for the p-series generalizes. Prove the so-called Cauchy condensation principle: $$ \sum_{n=1}^{\infty} x_{n} \quad \text { comverges if and only if } \quad \sum_{n=1}^{\infty} 2^{n} x_{2}^{n} \quad \text { converges. } $$

Let \(\left\\{x_{n}\right\\}\) be a sequence defined by $$x_{n}:=\left\\{\begin{array}{ll} n & \text { if } n \text { is odd } \\ 1 / n & \text { if } n \text { is even } \end{array}\right.$$ a) Is the sequence bounded? (prove or disprove) b) Is there a convergent subsequence? If so, find it.

Prove that \(\left\\{(n !)^{1 / n}\right\\}\) is unbounded. Hint: Show that \(\left\\{\frac{C^{n}}{n !}\right\\}\) converges to zero for any \(C>0\).

Suppose \(\left\\{x_{n}\right\\}\) is a bounded sequence, \(a_{n}:=\sup \left\\{x_{k}: k \geq n\right\\}\) as before. Suppose that for some \(\ell \in \mathbb{N}, a_{\ell} \notin\left\\{x_{k}: k \geq \ell\right\\} .\) Then show that \(a_{j}=a_{\ell}\) for all \(j \geq \ell\), and hence \(\limsup x_{n}=a_{\ell}\).

Let \(\left\\{x_{n}\right\\}\) be a sequence. Suppose there are two convergent subsequences \(\left\\{x_{n_{i}}\right\\}\) and \(\left\\{x_{m_{i}}\right\\} .\) Suppose $$\lim _{i \rightarrow \infty} x_{n_{i}}=a \quad \text { and } \quad \lim _{i \rightarrow \infty} x_{m_{i}}=b,$$ where \(a \neq b\). Prove that \(\left\\{x_{n}\right\\}\) is not convergent, without using Proposition 2.1.17.

Suppose \(\left\\{x_{n}\right\\}\) is a sequence, and \(a_{n}:=\sup \left\\{x_{k}: k \geq n\right\\}\) and \(b_{n}:=\sup \left\\{x_{k}: k \geq n\right\\}\) as before. a) Prove that if \(a_{\ell}=\infty\) for some \(\ell \in \mathbb{N},\) then \(\limsup x_{n}=\infty\). b) Prove that if \(b_{\ell}=-\infty\) for some \(\ell \in \mathbb{N},\) then \(\liminf x_{n}=-\infty\).

Prove Abel's theorem: Theorem. Suppose \(\sum x_{n}\) is a series whose partial sums are a bounded sequence, \(\left\\{\lambda_{n}\right\\}\) is a sequence with \(\lim \lambda_{n}=0,\) and \(\sum\left|\lambda_{n+1}-\lambda_{n}\right|\) is convergent. Then \(\sum \lambda_{n} x_{n}\) is convergent.

(Tricky): Find a sequence \(\left\\{x_{n}\right\\}\) such that for any \(y \in \mathbb{R},\) there exists a subsequence \(\left\\{x_{n_{i}}\right\\}\) converging to \(y .\)

Suppose \(\left\\{x_{n}\right\\}\) is a sequence such that both \(\liminf x_{n}\) and \(\limsup x_{n}\) are finite. Prove that \(\left\\{x_{n}\right\\}\) is bounded.

(Easy): Let \(\left\\{x_{n}\right\\}\) be a sequence and \(x \in \mathbb{R} .\) Suppose for any \(\varepsilon>0,\) there is an \(M\) such that for all \(n \geq M,\left|x_{n}-x\right| \leq \varepsilon .\) Show that \(\lim x_{n}=x\).

Suppose \(\left\\{x_{n}\right\\}\) is a bounded sequence, and \(\varepsilon>0\) is given. Prove that there exists an M such that for all \(k \geq M\) we have $$ x_{k}-\left(\limsup _{n \rightarrow \infty} x_{n}\right)<\varepsilon \quad \text { and } \quad\left(\liminf _{n \rightarrow \infty} x_{n}\right)-x_{k}<\varepsilon. $$

(Easy): Let \(\left\\{x_{n}\right\\}\) be a sequence and \(x \in \mathbb{R}\) such that there exists a \(k \in \mathbb{N}\) such that for all \(n \geq k, x_{n}=x .\) Prove that \(\left\\{x_{n}\right\\}\) converges to \(x .\)

Let \(\left\\{x_{n}\right\\}\) be a sequence and define a sequence \(\left\\{y_{n}\right\\}\) by \(y_{2 k}:=x_{k^{2}}\) and \(y_{2 k-1}:=x_{k}\) for all \(k \in \mathbb{N}\). Prove that \(\left\\{x_{n}\right\\}\) converges if and only if \(\left\\{y_{n}\right\\}\) converges. Furthermore, prove that if they converge, then \(\lim x_{n}=\lim y_{n}\).

Show that the 3 -tail of the sequence defined by \(x_{n}:=\frac{n}{n^{2}+16}\) is monotone decreasing. Hint: Suppose \(n \geq m \geq 4\) and consider the numerator of the expression \(x_{n}-x_{m}\).