Chapter 2

Decide the convergence or divergence of the following series. a) \(\sum_{n=1}^{\infty} \frac{1}{2^{2 n+1}}\) b) \(\sum_{n=1}^{\infty} \frac{(-1)^{n}(n-1)}{n}\) c) \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{1 / 10}}\) d) \(\sum_{n=1}^{\infty} \frac{n^{n}}{(n+1)^{2 n}}\)

Prove that \(\left\\{\frac{n^{2}-1}{n^{2}}\right\\}\) is Cauchy using directly the definition of Cauchy sequences.

Suppose the kth partial sum of \(\sum_{n=1} x_{n}\) is \(s_{k}=\frac{k}{k+1} .\) Find the series, that is find \(x_{n}\), prove that the series converges, and then find the limit.

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\\{3 n\\}\) bounded? Prove or disprove.

Suppose both \(\sum_{n=0}^{\infty} a_{n}\) and \(\sum_{n=0}^{\infty} b_{n}\) converge absolutely. Show that the product series, \(\sum_{n=0}^{\infty} c_{n}\) where \(c_{n}=a_{0} b_{n}+a_{1} b_{n-1}+\cdots+a_{n} b_{0},\) also converges absolutely.

Let \(\left\\{x_{n}\right\\}\) be a sequence such that there exists \(a 0

Prove that if \(\left\\{x_{n}\right\\}\) is a convergent sequence, \(k \in \mathbb{N}\), then \(\lim _{n \rightarrow \infty} x_{n}^{k}=\left(\lim _{n \rightarrow \infty} x_{n}\right)^{k}\) Hint: Use induction.

Suppose \(F\) is an ordered field that contains the rational numbers \(\mathbb{Q}\), such that \(\mathbb{Q}\) is dense, that is: Whenever \(x, y \in F\) are such that \(x0,\) there exists an \(M\) such that for all \(n, k \geq M\) we have \(\left|x_{n}-x_{k}\right|<\varepsilon\). Suppose any Cauchy sequence of rational numbers has a limit in \(F\). Prove that \(F\) has the least-upper-bound property.

Finish the proof of Proposition 2.3.6. That is, suppose \(\left\\{x_{n}\right\\}\) is a bounded sequence and \(\left\\{x_{n_{k}}\right\\}\) is a subsequence. Prove \(\liminf _{n \rightarrow \infty} x_{n} \leq \liminf _{k \rightarrow \infty} x_{n_{k}}\).

Decide the convergence or divergence of the following series. a) \(\sum_{n=1}^{\infty} \frac{3}{9 n+1}\) b) \(\sum_{n=1}^{\infty} \frac{1}{2 n-1}\) c) \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n^{2}}\) d) \(\sum_{n=1}^{\infty} \frac{1}{n(n+1)}\) e) \(\sum_{n=1}^{\infty} n e^{-n^{2}}\)

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{\frac{(-1)^{n}}{2 n}\right\\}\) convergent? If so, what is the limit?

Suppose \(x_{1}:=\frac{1}{2}\) and \(x_{n+1}:=x_{n}^{2}\). Show that \(\left\\{x_{n}\right\\}\) converges and find \(\lim x_{n}\). Hint: You cannot divide by zero!

a) Show that the alternating harmonic series \(\sum \frac{(-1)^{n-1}}{n}\) has a rearrangement such that for any \(x

Let \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) be sequences such that \(\lim y_{n}=0 .\) Suppose that for all \(k \in \mathbb{N}\) and for all \(m \geq k\) we have $$ \left|x_{m}-x_{k}\right| \leq y_{k} $$ Show that \(\left\\{x_{n}\right\\}\) is Cauchy.

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{2^{-n}\right\\}\) convergent? If so, what is the limit?

Let \(x_{n}:=\frac{n-\cos (n)}{n} .\) Use the squeeze lemma to show that \(\left\\{x_{n}\right\\}\) converges and find the limit.

For the following power series, find if they are convergent or not, and if so find their radius of convergence. a) \(\sum_{n=0}^{\infty} 2^{n} x^{n}\) b) \(\sum_{n=0}^{\infty} n x^{n}\) c) \(\sum_{n=0}^{\infty} n ! x^{n}\) d) \(\sum_{n=0}^{\infty} \frac{1}{(2 n) !}(x-10)^{n}\) e) \(\left.\sum_{n=0}^{\infty} x^{2 n} \quad f\right) \sum_{n=0}^{\infty} n ! x^{n !}\)

Suppose a Cauchy sequence \(\left\\{x_{n}\right\\}\) is such that for every \(M \in \mathbb{N},\) there exists a \(k \geq M\) and an \(n \geq M\) such that \(x_{k}<0\) and \(x_{n}>0 .\) Using simply the definition of a Cauchy sequence and of a convergent sequence, show that the sequence converges to \(0 .\)

a) Let \(x_{n}:=\frac{(-1)^{n}}{n} .\) Find \(\limsup x_{n}\) and \(\liminf x_{n}\). b) Let \(x_{n}:=\frac{(n-1)(-1)^{n}}{n} .\) Find \(\limsup x_{n}\) and \(\liminf x_{n}\).

For \(j=1,2, \ldots, n\), let \(\left\\{x_{j k}\right\\}_{k=1}^{\infty}\) denote n sequences. Suppose that for each \(j\) $$ \sum_{k=1}^{\infty} x_{j, k} $$ is convergent. Then show $$ \sum_{j=1}^{n}\left(\sum_{k=1}^{\infty} x_{j, k}\right)=\sum_{k=1}^{\infty}\left(\sum_{j=1}^{n} x_{j, k}\right) . $$

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{\frac{n}{n+1}\right\\}\) convergent? If so, what is the limit?

Suppose \(\sum a_{n} x^{n}\) converges for \(x=1\). a) What can you say about the radius of convergence? b) If you further know that at \(x=1\) the convergence is not absolute, what can you say?

Suppose \(\left|x_{n}-x_{k}\right| \leq n / k^{2}\) for all \(n\) and \(k .\) Show that \(\left\\{x_{n}\right\\}\) is Cauchy.

Let \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) be bounded sequences such that \(x_{n} \leq y_{n}\) for all \(n .\) Then show that $$ \limsup _{n \rightarrow \infty} x_{n} \leq \limsup _{n \rightarrow \infty} y_{n} $$ and $$ \liminf _{n \rightarrow \infty} x_{n} \leq \liminf _{n \rightarrow \infty} y_{n}. $$

Prove the following stronger version of the ratio test: Let \(\sum x_{n}\) be a series. a) If there is an \(N\) and a \(\rho<1\) such that for all \(n \geq N\) we have \(\frac{x_{n+1}}{x_{n}}<\rho,\) then the series converges absolutely. b) If there is an \(N\) such that for all \(n \geq N\) we have \(\frac{\left|x_{n+1}\right|}{x_{n}} \geq 1,\) then the series diverges.

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{\frac{n}{n^{2}+1}\right\\}\) convergent? If so, what is the limit?

True or false, prove or find a counterexample. If \(\left\\{x_{n}\right\\}\) is a sequence such that \(\left\\{x_{n}^{2}\right\\}\) converges, then \(\left\\{x_{n}\right\\}\) converges.

Expand \(\frac{x}{4-x^{2}}\) as a power series around \(x_{0}=0\) and compute its radius of convergence.

Suppose \(\left\\{x_{n}\right\\}\) is a Cauchy sequence such that for infinitely many \(n, x_{n}=c .\) Using only the definition of Cauchy sequence prove that \(\lim x_{n}=c .\)

Let \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) be bounded sequences. a) Show that \(\left\\{x_{n}+y_{n}\right\\}\) is bounded. b) Show that $$ \left(\liminf _{n \rightarrow \infty} x_{n}\right)+\left(\liminf _{n \rightarrow \infty} y_{n}\right) \leq \liminf _{n \rightarrow \infty}\left(x_{n}+y_{n}\right). $$ Hint: Find a subsequence \(\left\\{x_{n_{i}}+y_{n_{i}}\right\\}\) of \(\left\\{x_{n}+y_{n}\right\\}\) that converges. Then find a subsequence \(\left\\{x_{n_{m_{i}}}\right\\}\) of \(\left\\{x_{n_{i}}\right\\}\) that converges. Then apply what you know about limits. c) Find an explicit \(\left\\{x_{n}\right\\}\) and \(\left\\{y_{n}\right\\}\) such that $$ \left(\liminf _{n \rightarrow \infty} x_{n}\right)+\left(\liminf _{n \rightarrow \infty} y_{n}\right)<\liminf _{n \rightarrow \infty}\left(x_{n}+y_{n}\right). $$ Hint: Look for examples that do not have a limit.

Let \(\left\\{x_{n}\right\\}\) be a decreasing sequence such that \(\sum x_{n}\) converges. Show that \(\lim _{n \rightarrow \infty} n x_{n}=0\)

Show that $$ \lim _{n \rightarrow \infty} \frac{n^{2}}{2^{n}}=0 $$

a) Find an example where the radius of convergence of \(\sum a_{n} x^{n}\) and \(\sum b_{n} x^{n}\) are \(1,\) but the radius of convergence of the sum of the two series is infinite. b) (Trickier) Find an example where the radius of convergence of \(\sum a_{n} x^{n}\) and \(\sum b_{n} x^{n}\) are \(1,\) but the radius of convergence of the product of the two series is infinite.

True or false, prove or find a counterexample: If \(\left\\{x_{n}\right\\}\) is a Cauchy sequence, then there exists an \(M\) such that for all \(n \geq M\) we have \(\left|x_{n+1}-x_{n}\right| \leq\left|x_{n}-x_{n-1}\right| .\)

Show that \(\sum_{n=1}^{\infty} \frac{(-1)^{n}}{n}\) converges. Hint: Consider the sum of two subsequent entries.

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Is the sequence \(\left\\{\frac{2^{n}}{n !}\right\\}\) convergent? If so, what is the limit?

Suppose \(\left\\{x_{n}\right\\}\) is a sequence and suppose for some \(x \in \mathbb{R},\) the limit $$ L:=\lim _{n \rightarrow \infty} \frac{\left|x_{n+1}-x\right|}{\left|x_{n}-x\right|} $$ exists and \(L<1 .\) Show that \(\left\\{x_{n}\right\\}\) converges to \(x\).

If \(S \subset \mathbb{R}\) is a set, then \(x \in \mathbb{R}\) is a cluster point iffor every \(\varepsilon>0,\) the set \((x-\varepsilon, x+\varepsilon) \cap S \backslash\\{x\\}\) is not empty. That is, if there are points of \(S\) arbitrarily close to \(x .\) For example, \(S:=\\{1 / n: n \in \mathbb{N}\\}\) has \(a\) unique (only one) cluster point \(0,\) but \(0 \notin S .\) Prove the following version of the Bolzano-Weierstrass theorem: \mathrm{\\{} T h e o r e m . ~ L e t ~ \(S \subset \mathbb{R}\) be a bounded infinite set, then there exists at least one cluster point of \(S\). Hint: If \(S\) is infinite, then \(S\) contains a countably infinite subset. That is, there is a sequence \(\left\\{x_{n}\right\\}\) of distinct numbers in \(S .\)

a) Prove that if \(\sum x_{n}\) and \(\sum y_{n}\) converge absolutely, then \(\sum x_{n} y_{n}\) converges absolutely. b) Find an explicit example where the converse does not hold. c) Find an explicit example where all three series are absolutely comvergent, are not just finite sums, and \(\left(\sum x_{n}\right)\left(\sum y_{n}\right) \neq \sum x_{n} y_{n} .\) That is, show that series are not multiplied term-by- term.

In the following exercises, feel free to use what you know from calculus to find the limit, if it exists. But you must prove that you found the correct limit, or prove that the series is divergent. Show that the sequence \(\left\\{\frac{1}{\sqrt[3]{n}}\right\\}\) is monotone, bounded, and use Proposition 2.1.10 to find the limit.

(Challenging): Let \(\left\\{x_{n}\right\\}\) be a convergent sequence such that \(x_{n} \geq 0\) and \(k \in \mathbb{N}\). Then $$ \lim _{n \rightarrow \infty} x_{n}^{1 / k}=\left(\lim _{n \rightarrow \infty} x_{n}\right)^{1 / k} $$ Hint: Find an expression \(q\) such that \(\frac{x_{n}^{1 / k}-x^{1 / k}}{x_{n}-x}=\frac{1}{q} .\)

a) Prove that \(\lim n^{1 / n}=1 .\) Hint: Write \(n^{1 / n}=1+b_{n}\) and note \(b_{n}>0 .\) Then show that \(\left(1+b_{n}\right)^{n} \geq \frac{n(n-1)}{2} b_{n}^{2}\) and use this to show that \(\lim b_{n}=0 .\) b) Use the result of part a) to show that if \(\sum a_{n} x^{n}\) is a convergent power series with radius of convergence \(R\), then \(\sum n a_{n} x^{n}\) is also convergent with the same radius of convergence.

Let \(r>0 .\) Show that starting with any \(x_{1} \neq 0,\) the sequence defined by $$ x_{n+1}:=x_{n}-\frac{x_{n}^{2}-r}{2 x_{n}} $$ converges to \(\sqrt{r}\) if \(x_{1}>0\) and \(-\sqrt{r}\) if \(x_{1}<0\)

a) If \(\sum a_{n}\) is convergent to a (in the usual sense), show that \(\sum a_{n}\) is Cesàro summable (see above) to a. b) Show that in the sense of Cesàro \(\sum(-1)^{n}\) is summable to \(1 / 2\). c) Let \(a_{n}:=k\) when \(n=k^{3}\) for some \(k \in \mathbb{N}, a_{n}:=-k\) when \(n=k^{3}+1\) for some \(k \in \mathbb{N},\) otherwise let \(a_{n}:=0 .\) Show that \(\sum a_{n}\) diverges in the usual sense, (partial sums are unbounded), but it is Cesàro summable to 0 (seems a little paradoxical at first sight).

Prove the limit comparison test. That is, prove that if \(a_{n}>0\) and \(b_{n}>0\) for all \(n,\) and $$ 0<\lim _{n \rightarrow \infty} \frac{a_{n}}{b_{n}}<\infty $$ then either \(\sum a_{n}\) and \(\sum b_{n}\) both converge or both diverge.

a) Suppose \(\left\\{a_{n}\right\\}\) is a bounded sequence and \(\left\\{b_{n}\right\\}\) is a sequence converging to \(0 .\) Show that \(\left\\{a_{n} b_{n}\right\\}\) converges to 0 b) Find an example where \(\left\\{a_{n}\right\\}\) is unbounded, \(\left\\{b_{n}\right\\}\) converges to \(0,\) and \(\left\\{a_{n} b_{n}\right\\}\) is not convergent. c) Find an example where \(\left\\{a_{n}\right\\}\) is bounded, \(\left\\{b_{n}\right\\}\) converges to some \(x \neq 0,\) and \(\left\\{a_{n} b_{n}\right\\}\) is not convergent.

(Challenging): Show that the monotonicity in the alternating series test is necessary. That is, find a sequence of positive real numbers \(\left\\{x_{n}\right\\}\) with \(\lim x_{n}=0\) but such that \(\sum(-1)^{n} x_{n}\) diverges.

Let \(\left\\{x_{n}\right\\}\) be a bounded sequence. a) Prove that there exists an s such that for any \(r>s\) there exists an \(M \in \mathbb{N}\) such that for all \(n \geq M\) we have \(x_{n}

Let \(x_{n}=\sum_{j=1}^{n} 1 / j .\) Show that for every \(k\) we have \(\lim \left|x_{n+k}-x_{n}\right|=0,\) yet \(\left\\{x_{n}\right\\}\) is not Cauchy.

(Easy): Prove the following stronger version of Lemma 2.2.12, the ratio test. Suppose \(\left\\{x_{n}\right\\}\) is a sequence such that \(x_{n} \neq 0\) for all \(n\). a) Prove that if there exists an \(r<1\) and \(M \in \mathbb{N}\) such that for all \(n \geq M\) we have $$ \frac{\left|x_{n+1}\right|}{\left|x_{n}\right|} \leq r $$ then \(\left\\{x_{n}\right\\}\) converges to \(0 .\) b) Prove that if there exists an \(r>1\) and \(M \in \mathbb{N}\) such that for all \(n \geq M\) we have $$ \frac{\left|x_{n+1}\right|}{\left|x_{n}\right|} \geq r $$ then \(\left\\{x_{n}\right\\}\) is unbounded.