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Math 2.0

Special functions Proof

Proof of the exponential of a sum by the Cauchy product

Builds on Exponential function The exponential series has infinite radius of convergence Cauchy product Cauchy product of absolutely convergent series Binomial theorem Factorial formula for binomial coefficients

Proof

We multiply the two series by the Cauchy product and use the binomial theorem.

Let $x, y$ be real. Let $a_k = \frac{x^k}{k!}$ and $b_k = \frac{y^k}{k!}$ for $k \ge 0$ . By the definition of the exponential function, $\sum a_k = \exp(x)$ and $\sum b_k = \exp(y)$ .
Both series converge absolutely, since the exponential series has infinite radius of convergence.
The Cauchy product of the two series has the terms $c_n = \sum_{k=0}^{n} a_k b_{n-k} = \sum_{k=0}^{n} \frac{x^k y^{n-k}}{k! \, (n-k)!}$ .
By the factorial formula for binomial coefficients, $\frac{1}{k! \, (n-k)!} = \frac{1}{n!} \binom{n}{k}$ for $0 \le k \le n$ .
By step 4 and the binomial theorem in $\R$ , $c_n = \frac{1}{n!} \sum_{k=0}^{n} \binom{n}{k} x^k y^{n-k} = \frac{(x + y)^n}{n!}$ .
By step 2 and the theorem on the Cauchy product of absolutely convergent series, $\sum c_n$ converges, and its sum is $\exp(x) \exp(y)$ .
By step 5 and the definition of $\exp$ , $\sum_{n=0}^{\infty} c_n = \sum_{n=0}^{\infty} \frac{(x+y)^n}{n!} = \exp(x + y)$ .
Comparing steps 6 and 7 gives $\exp(x + y) = \exp(x) \exp(y)$ .

Idea

Multiplying the two series and collecting the terms of total degree $n$ gives exactly the binomial expansion of $(x + y)^n$ , divided by $n!$ . Absolute convergence makes this rearrangement legitimate.

Reading path

To understand this page from the ground up, read these 157 pages in order. The foundations come first. The path takes the proofs Proof of the Weierstrass M-test, Proof of the limit of a sum of sequences, Proof of the approximation property of the supremum, Proof that every sequence has a monotone subsequence, by peaks, Proof of the commutativity of addition by induction, Proof that absolutely convergent series converge, Proof of the sum of a finite geometric series, Proof that convergent sequences are Cauchy, Proof of the Cauchy criterion for series, Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem, Proof that the exponential series has infinite radius of convergence, Proof that the reals form an ordered field, with the Cauchy sequence construction, Proof of the double negation law by a truth table, Proof that power series converge uniformly on smaller intervals inside the radius of convergence, Proof of strong induction from ordinary induction, Proof of the limit of a product of sequences, Proof that the absolute value is multiplicative, Proof of the squeeze theorem for sequences, Proof of Pascal's rule from the recursive definition, Proof that Cauchy sequences are bounded, Proof of the triangle inequality for real numbers, Proof of the epsilon-N definition of the limit of a sequence, Proof of the comparison test, Proof that limits preserve non-strict inequalities, Proof of the factorial formula for binomial coefficients by induction, Proof of the binomial theorem by induction, Proof of the well-ordering principle from strong induction, Proof of the Bolzano–Weierstrass theorem by a monotone subsequence, Proof of the Archimedean property in an Archimedean ordered field, Proof that a decreasing sequence bounded below converges, Proof that the reals are Archimedean from the completeness axiom, Proof of the ratio test, Proof that convergent sequences are bounded, Proof of the monotone convergence theorem for sequences, Proof of the ratio formula for the radius of convergence, Proof that a series of nonnegative terms converges iff its partial sums are bounded, Proof that the Cauchy product of absolutely convergent series converges absolutely to the product.

Binomial coefficient definition
Construction of the reals from Cauchy sequences concept
Definition by recursion concept
Element definition
Factorial definition
Group concept
Integers object
Mathematical induction concept
Natural numbers object
Ordered field definition
Peano axioms concept
Proposition concept
Rational numbers object
Real numbers object
Sequence definition
Set bounded above definition
Proof of Pascal's rule from the recursive definition proof
Set definition
Ring definition
Ring of integers statement
Archimedean ordered field definition
Successor function definition
Conjunction definition
Disjunction definition
Implication definition
Negation definition
Predicate definition
The reals form a field statement
Real sequence definition
Subsequence definition
Infimum definition
Supremum definition
Pascal's rule statement
Ordered pair definition
Commutative ring definition
Addition of natural numbers definition
Proof of the Archimedean property in an Archimedean ordered field proof
Biconditional definition
The rationals form a field statement
The successor function is injective statement
Truth table concept
Zero is not a successor statement
Modus ponens concept
Universal quantifier definition
Bounded sequence definition
Cauchy sequence definition
Geometric sequence definition
Limit of a sequence definition
Monotone sequence definition
Series definition
Completeness axiom statement
Cartesian product definition
Proof of the factorial formula for binomial coefficients by induction proof
Archimedean property statement
Proof of the commutativity of addition by induction proof
Order on the natural numbers definition
Subset definition
The rationals form an ordered field statement
Propositional logic concept
Mathematical proof concept
Proof of the sum of a finite geometric series proof
Convergent sequence definition
Pointwise convergence definition
Cauchy product definition
Partial sum definition
Power series definition
Factorial formula for binomial coefficients statement
Binary relation definition
Order on the reals definition
Commutativity of addition statement
Proof of strong induction from ordinary induction proof
Tautology definition
Uniform convergence definition
Sum of a finite geometric series statement
Radius of convergence definition
Convergent series definition
Proof of the binomial theorem by induction proof
Antisymmetric relation definition
Reflexive relation definition
Transitive relation definition
Strong induction statement
Logical equivalence definition
Uniform convergence on a set definition
Sum of a series definition
Series of functions definition
Absolute convergence definition
Binomial theorem statement
Preorder definition
Proof of the well-ordering principle from strong induction proof
Proof of the double negation law by a truth table proof
Partial order definition
Well-ordering principle statement
Double negation law statement
Total order definition
Proof by contradiction concept
Proof that every sequence has a monotone subsequence, by peaks proof
Proof that the reals form an ordered field, with the Cauchy sequence construction proof
Every sequence has a monotone subsequence statement
The reals form an ordered field statement
Absolute value definition
Proof of the approximation property of the supremum proof
Proof of the triangle inequality for real numbers proof
Proof that the absolute value is multiplicative proof
Proof of the epsilon-N definition of the limit of a sequence proof
Approximation property of the supremum statement
Triangle inequality statement
Absolute value is multiplicative statement
Epsilon-N definition of the limit of a sequence statement
Proof that the reals are Archimedean from the completeness axiom proof
Proof of the limit of a sum of sequences proof
Proof of the squeeze theorem for sequences proof
Proof that limits preserve non-strict inequalities proof
Proof that convergent sequences are bounded proof
Proof that convergent sequences are Cauchy proof
The reals are Archimedean statement
Proof that Cauchy sequences are bounded proof
Proof of the monotone convergence theorem for sequences proof
Limit of a sum of sequences statement
Limits preserve non-strict inequalities statement
Convergent sequences are bounded statement
Proof of the ratio formula for the radius of convergence proof
Squeeze theorem statement
Convergent sequences are Cauchy statement
Cauchy sequences are bounded statement
Monotone convergence theorem for sequences statement
Proof of the limit of a product of sequences proof
Proof that a series of nonnegative terms converges iff its partial sums are bounded proof
Proof that a decreasing sequence bounded below converges proof
The sequence r^n tends to zero for |r| < 1 example
Decreasing sequence bounded below converges statement
Limit of a product of sequences statement
A series of nonnegative terms converges iff its partial sums are bounded statement
Geometric series example
Proof that the Cauchy product of absolutely convergent series converges absolutely to the product proof
Proof of the ratio test proof
Proof of the comparison test proof
Proof of the Bolzano–Weierstrass theorem by a monotone subsequence proof
Comparison test statement
Cauchy product of absolutely convergent series statement
Bolzano–Weierstrass theorem statement
Ratio test statement
Proof of the Cauchy criterion for sequences by the Bolzano–Weierstrass theorem proof
Ratio formula for the radius of convergence statement
Cauchy criterion for sequences statement
Uniformly Cauchy sequence of functions definition
Proof of the Cauchy criterion for series proof
Cauchy criterion for series statement
Proof that absolutely convergent series converge proof
Absolutely convergent series converge statement
Proof of the Weierstrass M-test proof
Weierstrass M-test statement
Proof that power series converge uniformly on smaller intervals inside the radius of convergence proof
Power series converge uniformly on smaller discs inside the radius of convergence statement
Proof that the exponential series has infinite radius of convergence proof
The exponential series has infinite radius of convergence statement
Exponential function definition
Proof of the exponential of a sum by the Cauchy product proof

Where this leads

325 pages build on this one.

Further reading

Web https://en.wikipedia.org/wiki/Cauchy_product, Examples

About this page

Tag	`S5QC84WQPW`
URI	`m2://S5QC84WQPW`
Kind	proof
Writing	W0 stub
MSC	33B10
Proves	Exponential of a sum
Uses	Exponential function, The exponential series has infinite radius of convergence, Cauchy product, Cauchy product of absolutely convergent series, Binomial theorem, Factorial formula for binomial coefficients
Changed	4 November 2026

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