Exercise 3.15

For $a_n\in {ℝ}^k$, the series $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges if and only if for every $𝜖>0$ there is an integer $N$ such that

for all $m\ge n\ge N$.

Proof: $\text{(→)}$ Suppose that $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges. Then by definition the sequence $\{s_n\}$ of partial sums converges where $s_n={\sum <!--\; FUNCTION\; APPLICATION\; -->}_{k=1}^n{a}_k$. It follows from Theorem 3.11a that $\{s_n\}$ is a Cauchy sequence, noting that this is true in any metric space. So consider any $𝜖>0$. Then, since $\{s_n\}$ is Cauchy, there is an integer $N$ such that for every $m\ge N$ and $n\ge N$, $|s_m-s_n|<𝜖$.

So let $M=N+1$ and consider any $m\ge n\ge M$. Also let $l=n-1$, from which it follows immdediately that $n=l+1$. Clearly then $m\ge l\ge N$ so that $|s_m-s_l|<𝜖$. We then have

as required.

$\text{(←)}$ Suppose that for every $𝜖>0$ there is an integer $N$ such that

for all $m\ge n\ge N$ and consider any $𝜖>0$. Then obviously there is an integer $N$ for which

for every $m\ge n\ge N$. So consider any $m\ge N$ and $n\ge N$. We can assume that $m\ge n$ without loss of generality. If $m=n$ then we have simply $|s_m-s_n|=|s_m-s_m|=|0|=0<𝜖$. If $m>n$ let $l=n+1$. It then follows that $m\ge l\ge N$ so that

We then have

Thus we have shown that $\{s_n\}$ is a Cauchy sequence. Then by Theorem 3.11c, $\{s_n\}$ converges since we are in ${ℝ}^k$. It follows then by definition that $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges.

For $a_n\in {ℝ}^k$, if $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges then $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{a}_n=0$.

Proof: Suppose that $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges and consider any $𝜖>0$. Then by Theorem 3.22 there is an integer $N$ such that

for all $m\ge n\ge N$. So consider any $n\ge N$ and let $m=n$. Then clearly $m\ge n\ge N$ is true so that

thereby showing by definition that $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{a}_n=0$.

For $a_n\in {ℝ}^k$ and $c_n\in ℝ$, if $|a_n|\le c_n$ for all $n\ge N_0$ where $N_0$ is a fixed integer, and if $\sum <!--\; FUNCTION\; APPLICATION\; -->c_n$ converges, then $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges.

Proof: The proof is the same as that in the text since Theorem 3.22 applies to sums in ${ℝ}^k$ and the triangle inequality is also true there.

For $a_n\in {ℝ}^k$, given $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$, put $\alpha ={\mathrm{limsup}<!--\; FUNCTION\; APPLICATION\; -->}_{n\to \infty }\sqrt[n]{|a_n|}$. Then

(a) if $\alpha <1$ then $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges
(b) if $\alpha >1$ then $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ diverges
(c) if $\alpha =1$, then the test gives no information.

Proof: The proof is again the same as that given in the text since for each part:

(a) The comparison test (Theorem 3.25a) is valid in ${ℝ}^k$.
(b) Theorem 3.23 is valid in ${ℝ}^k$.
(c) The series given still provide a counter example since they are in ${ℝ}^1$

For $a_n\in {ℝ}^k$, the series $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$

(a) converges if ${\mathrm{limsup}<!--\; FUNCTION\; APPLICATION\; -->}_{n\to \infty }\left|\frac{a_{n+1}}{a_n}\right|<1$
(b) diverges if $\left|\frac{a_{n+1}}{a_n}\right|\ge 1$ for all $n\ge N_0$, where $N_0$ is some fixed integer.

Proof: Similar to the root test, the proof is the same as that given in the text since for each part:

(a) The comparison test (Theorem 3.25a) is valid in ${ℝ}^k$.
(b) Theorem 3.23 is valid in ${ℝ}^k$.

For $a_n\in {ℝ}^k$ and $b_n\in ℝ$ suppose that

(a) the partial sums $A_n$ of $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ form a bounded sequence;
(b) $b_0\ge b_1\ge b_2\ge \cdots$;
(c) $\underset{n\to \infty }{\lim <!--\; FUNCTION\; APPLICATION\; -->}{b}_n=0$.

Then $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n{b}_n$ converges.

Proof: The proof given in the text still holds since Theorem 3.41 clearly holds for $a_n\in {ℝ}^k$ as does the Cauchy criterion (Theorem 3.22).

For $a_n\in {ℝ}^k$, if $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges absolutely, then $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges.

Proof: The proof is the same since the Cauchy criterion (Theorem 3.22) and the triangle inequality both hold in ${ℝ}^k$.

For $a_n\in {ℝ}^k$ and $b_n\in {ℝ}^k$, if $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n=A$ and $\sum <!--\; FUNCTION\; APPLICATION\; -->b_n=B$, then $\sum <!--\; FUNCTION\; APPLICATION\; -->(a_n+b_n)=A+B$, and $\sum <!--\; FUNCTION\; APPLICATION\; -->c{a}_n=\mathit{cA}$ for any fixed $c\in ℝ$.

Proof: The proofs are the same as those given in the text since the limit rules used (Theorem 3.3 parts a and b) are trivially shown to hold in ${ℝ}^k$.

If $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ is a series of elements in ${ℝ}^k$ that converges absolutely, then every rearrangement of $\sum <!--\; FUNCTION\; APPLICATION\; -->a_n$ converges, and they all converge to the same sum.

Proof: The proof is the same since the Cauchy criterion (Theorem 3.22) holds in ${ℝ}^k$.