Addition and strict positivity of the full exponential series

Goal and scope

From E(x)=sum x^n/n!, prove absolute convergence for all real x, addition, inverses and positivity.

No derivative, logarithm, ODE, injectivity or transcendence claim.

Givens and symbols

• algebra
• epsilon
• cauchy_product
• real_complete

• G1 — E(x) is the whole series x^n/n! over n≥0; x,y are arbitrary fixed real numbers.

• E(x) — whole power-series function
• u_n — absolute term
• m — arbitrary total degree

Rules and sources

Some rule statements have reader clarifications in this revision: they are paraphrases, not verbatim quotations. Referenced original source bytes and hashes are unchanged.

• mko_definition · cited — Terms at x are x^n/n!, while the power-series coefficients 1/n! have ratio1/(n+1) tending to 0, giving infinite radius and absolute convergence for every fixed real x. The addition law needs the binomial coefficient identity at every degree and absolute convergence for Cauchy-product rearrangement; one declared degree checks only one coefficient. E(0)=1 and E(x)E(−x)=1. The half inverse identity makes E(x/2) nonzero before E(x)=E(x/2)^2 proves strict positivity. A finite partial-sum computation is not the whole exponential function, even where a special value coincides. Derivatives, logarithms, ODE uniqueness, injectivity and exact rational values at nonzero rational inputs are not automatic claims of this node. · mko-real-exponential-function
• algebra · prerequisite — Use real-field identities; division requires a nonzero denominator and multiplication by a positive number preserves inequalities.
• epsilon · prerequisite — Convergence requires for every positive epsilon an integer N controlling every n≥N by a strict error bound; use the Archimedean property of the reals.
• cauchy_product · prerequisite — The product of two absolutely convergent series equals their Cauchy product and may be grouped by total degree; the binomial theorem holds for every nonnegative integer degree.
• real_complete · prerequisite — Every nonempty bounded-above real set has a supremum, so bounded monotone real sequences converge, including bounded partial sums of nonnegative series.

Main derivation

u_n=|x|^n/n!,\quad x\ne0

u_{n+1}/u_n=|x|/(n+1)

N>2|x|,\quad n\ge N

u_{n+1}\le\tfrac12u_n,\quad u_{N+j}\le2^{-j}u_N

\sum_{j=0}^{M}u_{N+j}\le2u_N

\sum_{n=0}^{\infty}|x|^n/n!<\infty\quad\forall x\in\mathbb R

\sum_{k=0}^{m}\frac{x^ky^{m-k}}{k!(m-k)!}

\frac1{m!}\sum_{k=0}^{m}\binom mkx^ky^{m-k}=\frac{(x+y)^m}{m!}

E(x)E(y)=\sum_{m=0}^{\infty}\sum_{k=0}^{m}\frac{x^ky^{m-k}}{k!(m-k)!}

E(x)E(y)=E(x+y)

E(0)=1,\quad y=-x

E(x)E(-x)=1

E(x/2)E(-x/2)=1

E(x/2)\ne0

E(x)=E(x/2+x/2)=E(x/2)^2

E(x)>0\quad\forall x\in\mathbb R

Substeps and lemmas

No substeps in this guide.

Reverse audits

• S08 · premise_audit · S05, S07 — No finite partial sum at a nonzero rational point has been called exact E(x).
• S07 · premise_audit · S06 — A nonnegative square alone does not establish strict positivity.
• S06 · premise_audit · G1, S05 — Finite partial sums generally do not have this exact inverse property.
• S05 · premise_audit · S03, S04 — The coefficient identity and infinite regrouping are separate obligations, both supplied.
• S04 · inverse_equivalence · G1, cauchy_product — The factorial denominators are valid because 0≤k≤m.
• S03 · premise_audit · S02, real_complete — This step assumes real bounded-monotone convergence, not sampled finite M values.
• S02 · premise_audit · S01 — N may depend on the fixed x, but the bound covers every j.
• S01 · premise_audit · G1 — No ratio division by a zero term is performed at x=0.

Error controls and negative controls

Whole series

Separate the full E from finite S_N.

Nonzero first

Use inverses to exclude zero before deriving positivity from a square.

• A finite partial sum is the general exact E(x). → reject — It omits an infinite tail; equality requires a separately proved special value.
• Checking degree three alone proves addition. → reject — Every degree identity and absolute regrouping are required.
• The square identity alone proves strict positivity. → reject — E(x/2)=0 must first be excluded.

Result and limitations

E converges absolutely on R and satisfies addition, E(0)=1, E(x)E(-x)=1 and E(x)>0.

Verification and source binding

1 declared checks for this guide; only the listed expressions are covered.

• source commit: b1b2f1dcf8c50d67a93fe52d95f8eef7b3abba9f
• adopted source SHA-256: a1687161d95b6c66cce720bd37cdfa330f5a9bfd717247f0fa87c5f638f6ca35
• original source SHA-256: ea4e23ef55650baa4c70e04bfc52f0293ed62cfd1b2f52bb15cc8695fe035c75
• Lean: NOT_RUN · independent AI: SCOPED_LEAD_AND_BILINGUAL_CONTEXT_REVIEW

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