Proof of Series of Complex Numbers: Absolute Convergence, Comparison, Linearity, Index Shifts, Real and Imaginary Parts, Conjugates, Real Series and the Cauchy Product

Routine facts are used without comment (Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity); among them, that termwise formulas such as $(\operatorname{Re}a_{k})$, $(|a_{k}|)$, $(a_{k}+b_{k})$ or $\big(\sum_{k=1}^{n}a_{k}b_{n+1-k}\big)_{n}$ define sequences. The results on real sequences and series cited below are read in the current definitions of sequences, convergence, limits and series (The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (i), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (ii), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (iii), The Earlier and the New Versions of the Definitions of Sequences, Limits, Cauchy Sequences, Series and Iterated Operations Agree (v)), and the results on conjugates and moduli for the current complex numbers, conjugate and modulus (The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (ii), The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iii), The Earlier and the New Versions of the Definitions of the Natural Numbers, the Complex Numbers, the Conjugate and the Modulus Agree, and the Two Versions of Real Correspond (iv)). Write $s_{n}$ and $t_{n}$ for the $n$-th partial sums of $(a_{k})$ and $(b_{k})$, and $A$, $B$ for the sums of the two series when they exist.

Step 1: finite sums, part by part. Real and imaginary parts, conjugation and the modulus behave on finite sums as Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (viii) says: for every $n\in\mathbb{N}$,

$$\operatorname{Re}s_{n}=\sum_{k=1}^{n}\operatorname{Re}a_{k},\qquad\operatorname{Im}s_{n}=\sum_{k=1}^{n}\operatorname{Im}a_{k},\qquad\overline{s_{n}}=\sum_{k=1}^{n}\overline{a_{k}},\qquad|s_{n}|\le\sum_{k=1}^{n}|a_{k}|.$$

So the partial sums of the real series $\sum_{k}\operatorname{Re}a_{k}$ and $\sum_{k}\operatorname{Im}a_{k}$ are the real and imaginary parts of the $s_{n}$, and those of $\sum_{k}\overline{a_{k}}$ are the $\overline{s_{n}}$. The same identities and inequality hold for every finite sum of complex numbers, and they are used in that form below.

Step 2: real and imaginary parts, and conjugates. We prove (v) and (vi). By Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (ii), $s_{n}\to w$ holds exactly when $\operatorname{Re}s_{n}\to\operatorname{Re}w$ and $\operatorname{Im}s_{n}\to\operatorname{Im}w$. If $\sum_{k}a_{k}$ converges with sum $A$, then $s_{n}\to A$ (The Limit of a Convergent Complex Sequence (i)), so by Step 1 the partial sums of $\sum_{k}\operatorname{Re}a_{k}$ and $\sum_{k}\operatorname{Im}a_{k}$ converge to $\operatorname{Re}A$ and $\operatorname{Im}A$: the two real series converge, with these sums (The Limit of a Convergent Sequence (i)). Conversely, if the partial sums of the two real series tend to $\alpha$ and $\beta$, then $w=\alpha+\beta i$ has real part $\alpha$ and imaginary part $\beta$ (Real and Imaginary Parts of a Complex Number (i)), so $s_{n}\to\alpha+\beta i$ and $\sum_{k}a_{k}$ converges. This is (v). For (vi): if $s_{n}\to A$, then $\overline{s_{n}}\to\overline{A}$ by Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (v), and $\overline{s_{n}}$ is the $n$-th partial sum of $\sum_{k}\overline{a_{k}}$ by Step 1; so that series converges, and its sum is $\overline{A}$ (Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (ii)).

Step 3: real series. We prove (vii). Each $\iota(r_{k})$ is $r_{k}+0i$, so $\operatorname{Re}\iota(r_{k})=r_{k}$ and $\operatorname{Im}\iota(r_{k})=0$, by the uniqueness in The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii) and Real and Imaginary Parts of a Complex Number (i). The series of imaginary parts is therefore the series of zeros; its partial sums are all $0$, so it converges with sum $0$ (Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (i)). By Step 2, $\sum_{k}\iota(r_{k})$ converges exactly when $\sum_{k}r_{k}$ does, and then its sum $S$ has $\operatorname{Re}S=\sum_{k}r_{k}$ and $\operatorname{Im}S=0$. Hence $S=\big(\sum_{k}r_{k}\big)+0i=\iota\big(\sum_{k}r_{k}\big)$, by the uniqueness in The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii).

Step 4: linearity. We prove (iii). By Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (v) and Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (i), the $n$-th partial sums of $(a_{k}+b_{k})$ and $(\lambda a_{k})$ are

$$\sum_{k=1}^{n}(a_{k}+b_{k})=s_{n}+t_{n},\qquad\sum_{k=1}^{n}\lambda a_{k}=\lambda s_{n}.$$

If $s_{n}\to A$ and $t_{n}\to B$, then $s_{n}+t_{n}\to A+B$; and the constant sequence $(\lambda)$ converges to $\lambda$, so the product sequence $\lambda s_{n}$ converges to $\lambda A$. Both statements are Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (iii).

Step 5: index shifts. We prove (iv). For $p=0$ the two series coincide and the finite sum is $0$, so there is nothing to prove. Let $p\in\mathbb{N}$. By (v), applied to $(a_{k})$ and to $(a_{p+k})_{k}$, each of the two series converges exactly when the series of its real parts and of its imaginary parts converge; and by Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (iv), $\sum_{k}\operatorname{Re}a_{k}$ converges exactly when $\sum_{k}\operatorname{Re}a_{p+k}$ does, and likewise for imaginary parts. So the two complex series converge together. In that case, compare real parts: by Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (ii), Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (viii) for the finite sum, (v) for the two series and the sum formula of Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (iv),

$$\operatorname{Re}\Big(\sum_{k=1}^{p}a_{k}+\sum_{k=1}^{\infty}a_{p+k}\Big)=\sum_{k=1}^{p}\operatorname{Re}a_{k}+\sum_{k=1}^{\infty}\operatorname{Re}a_{p+k}=\sum_{k=1}^{\infty}\operatorname{Re}a_{k}=\operatorname{Re}\sum_{k=1}^{\infty}a_{k},$$

and in the same way for imaginary parts. Two complex numbers with the same real and imaginary parts are equal (Real and Imaginary Parts of a Complex Number (i)), which gives the displayed identity.

Step 6: absolute convergence. We prove (i). Suppose $\sum_{k}a_{k}$ converges absolutely, that is, $\sum_{k}|a_{k}|$ converges (Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii)). Since $|\operatorname{Re}a_{k}|\le|a_{k}|$ and $|\operatorname{Im}a_{k}|\le|a_{k}|$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (vi)), the real series $\sum_{k}\operatorname{Re}a_{k}$ and $\sum_{k}\operatorname{Im}a_{k}$ converge absolutely by Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (iv), hence converge by Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (iii); by Step 2, $\sum_{k}a_{k}$ converges, with sum $A$ say. For the inequality, Step 1 gives

$$|s_{n}|\le\sum_{k=1}^{n}|a_{k}|\qquad(n\in\mathbb{N}).$$

As $n\to\infty$ the left side tends to $|A|$, by Limits of Complex Sequences: Uniqueness, Real and Imaginary Parts, Arithmetic, Real Sequences, Conjugates, Moduli, the Cauchy Criterion and the Modulus Metric (v), and the right side to $\sum_{k}|a_{k}|$; so $|A|\le\sum_{k}|a_{k}|$ by Limits of Sequences of Real Numbers: Uniqueness, Boundedness, Constants, Tails, Arithmetic, Quotients, Absolute Values, Finite Sums, Order, Squeezing, Domination and Subsequences (x).

Step 7: comparison. We prove (ii). Suppose $|a_{k}|\le r_{k}$ for every $k$ and $\sum_{k}r_{k}$ converges. Then $0\le|a_{k}|\le r_{k}$ (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iii)), so $\sum_{k}|a_{k}|$ converges and $\sum_{k}|a_{k}|\le\sum_{k}r_{k}$, by Series of Real Numbers with Nonnegative Terms, Comparison, Absolute Convergence and the Geometric Series (ii). Thus $\sum_{k}a_{k}$ converges absolutely (Series of Complex Numbers: Partial Sums, Convergence, the Sum and Absolute Convergence (iii)), and by Step 6

$$\Big|\sum_{k=1}^{\infty}a_{k}\Big|\le\sum_{k=1}^{\infty}|a_{k}|\le\sum_{k=1}^{\infty}r_{k}.$$

The remaining steps prove (viii), by splitting $c_{n}$ into real and imaginary parts and applying the Cauchy product theorem for real series, The Cauchy Product of Two Absolutely Convergent Series (ii), to four pairs of real series.

Step 8: four real series. Put

$$\alpha_{k}=\operatorname{Re}a_{k},\qquad\beta_{k}=\operatorname{Im}a_{k},\qquad\gamma_{k}=\operatorname{Re}b_{k},\qquad\delta_{k}=\operatorname{Im}b_{k}.$$

As in Step 6, the four real series $\sum_{k}\alpha_{k}$, $\sum_{k}\beta_{k}$, $\sum_{k}\gamma_{k}$ and $\sum_{k}\delta_{k}$ converge absolutely. So do the real series $\sum_{k}|a_{k}|$ and $\sum_{k}|b_{k}|$: their terms are nonnegative (Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iii)), so they equal their absolute values, and the two series converge by hypothesis. Hence The Cauchy Product of Two Absolutely Convergent Series applies to any two of these six series.

Step 9: real and imaginary parts of $c_{n}$. By the product formula of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii), $\operatorname{Re}(a_{k}b_{n+1-k})=\alpha_{k}\gamma_{n+1-k}-\beta_{k}\delta_{n+1-k}$ and $\operatorname{Im}(a_{k}b_{n+1-k})=\alpha_{k}\delta_{n+1-k}+\beta_{k}\gamma_{n+1-k}$. Summing over $k$, Step 1 applied to the finite sum $c_{n}$, together with Finite Sums in a Commutative Ring and in an Ordered Field: Distributivity, Differences, Telescoping, Constant Terms, Comparison and the Triangle Inequality (ii) for the real part and Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (v) for the imaginary part, gives

$$\operatorname{Re}c_{n}=\sum_{k=1}^{n}\alpha_{k}\gamma_{n+1-k}-\sum_{k=1}^{n}\beta_{k}\delta_{n+1-k},\qquad\operatorname{Im}c_{n}=\sum_{k=1}^{n}\alpha_{k}\delta_{n+1-k}+\sum_{k=1}^{n}\beta_{k}\gamma_{n+1-k}.$$

The real Cauchy product of The Cauchy Product of Two Absolutely Convergent Series sums over the diagonal sets $D_{n}=\{(i,j)\in[n]\times[n]:i+j=n+1\}$, and the first of these four sums is $\sum_{(i,j)\in D_{n}}\alpha_{i}\gamma_{j}$, the $n$-th term of the real Cauchy product of $(\alpha_{k})$ and $(\gamma_{k})$ (rewriting the sums over $D_{n}$ as sums over $k$ by Iterated Operations over Finite Sets: Singletons, Disjoint Unions, Reindexing, Products of Sets, Termwise Combination, Homomorphisms and Intervals (viii) and The Diagonal Sets of the Cauchy Product and Re-indexing of Diagonal Sums (iv), the latter with $m=n-1$: for every real-valued map $g$ on $\mathbb{N}\times\mathbb{N}$, here $g(i,j)=\alpha_{i}\gamma_{j}$,

$$\sum_{k=1}^{n}g(k,n+1-k)=\sum_{k=0}^{n-1}g(k+1,n-k)=\sum_{(i,j)\in D_{n}}g(i,j),$$

since $n+1-(k+1)=n-k=((n-1)-k)+1$); by The Cauchy Product of Two Absolutely Convergent Series (ii) and Step 8, its series converges, with sum $\big(\sum_{k}\alpha_{k}\big)\big(\sum_{k}\gamma_{k}\big)$. The other three sums are treated in the same way. By Elementary Properties of Series of Real Numbers: Linearity, Null Terms, the Cauchy Criterion, Index Shifts, Tails, Order and Telescoping (i) (sums, and multiples by $-1$), the series $\sum_{n}\operatorname{Re}c_{n}$ and $\sum_{n}\operatorname{Im}c_{n}$ converge, with

$$\sum_{n=1}^{\infty}\operatorname{Re}c_{n}=\sum_{k=1}^{\infty}\alpha_{k}\sum_{k=1}^{\infty}\gamma_{k}-\sum_{k=1}^{\infty}\beta_{k}\sum_{k=1}^{\infty}\delta_{k},\qquad\sum_{n=1}^{\infty}\operatorname{Im}c_{n}=\sum_{k=1}^{\infty}\alpha_{k}\sum_{k=1}^{\infty}\delta_{k}+\sum_{k=1}^{\infty}\beta_{k}\sum_{k=1}^{\infty}\gamma_{k}.$$

Step 10: the sum of the Cauchy product. By Step 9 and Step 2, $\sum_{n}c_{n}$ converges, and its sum $C$ has the real and imaginary parts displayed in Step 9. The series $\sum_{k}a_{k}$ and $\sum_{k}b_{k}$ converge by Step 6, and by Step 2 their sums satisfy $\operatorname{Re}A=\sum_{k}\alpha_{k}$, $\operatorname{Im}A=\sum_{k}\beta_{k}$, $\operatorname{Re}B=\sum_{k}\gamma_{k}$ and $\operatorname{Im}B=\sum_{k}\delta_{k}$. The product formula of The Complex Numbers, with the Real Numbers Identified with a Subset of the Complex Numbers (vii) gives

$$AB=\big(\operatorname{Re}A\operatorname{Re}B-\operatorname{Im}A\operatorname{Im}B\big)+\big(\operatorname{Re}A\operatorname{Im}B+\operatorname{Im}A\operatorname{Re}B\big)i,$$

so $C$ and $AB$ have the same real and imaginary parts, and $C=AB$ (Real and Imaginary Parts of a Complex Number (i)).

Step 11: absolute convergence of the Cauchy product. For every $n$, the inequality of Step 1 for the finite sum $c_{n}$ and Conjugation and Modulus of Complex Numbers: Field Automorphism, Real and Imaginary Parts, Multiplicativity, Inverses, Component Bounds, Triangle Inequalities and Finite Sums (iv) give

$$|c_{n}|\le\sum_{k=1}^{n}|a_{k}|\,|b_{n+1-k}|.$$

By the rewriting in Step 9, now with $g(i,j)=|a_{i}|\,|b_{j}|$, the right side is the $n$-th term of the real Cauchy product of $(|a_{k}|)$ and $(|b_{k}|)$, whose series converges by The Cauchy Product of Two Absolutely Convergent Series (ii) and Step 8. By (ii), with this sequence for $(r_{n})$, the series $\sum_{n}c_{n}$ converges absolutely. Together with Step 10 this proves (viii).