= Solution
Work near the identity $O$ of a generalized <Weierstrass equation of an elliptic curve>. On the projective chart where $Y\ne0$, use the <local parameter> $t=-x/y$ and the auxiliary coordinate $w=-1/y$. Substitution gives
$$
w=t^3+a_1tw+a_2t^2w+a_3w^2+a_4tw^2+a_6w^3.
$$
The coefficient of $w$ in the linearized equation at $(0,0)$ is one. Consequently, successive coefficient comparison determines a unique <formal power series> $w(t)$, without dividing by any coefficient of the equation. Its initial terms are
$$
w(t)=t^3+a_1t^4+(a_1^2+a_2)t^5+(a_1^3+2a_1a_2+a_3)t^6+O(t^7).
$$
Recover the affine coordinates as Laurent series $x=t/w(t)$ and $y=-1/w(t)$; in particular $x=t^{-2}+O(t^{-1})$ and $y=-t^{-3}+O(t^{-2})$.
The <elliptic curve group law> is a regular morphism near $(O,O)$. Applying the local parameter to the sum and passing to completed local rings produces
$$
F(T_1,T_2)=t(P+Q)\in k[[T_1,T_2]].
$$
One can construct the series without any nonunit division. Put
$$
\lambda=\frac{w(T_2)-w(T_1)}{T_2-T_1},\qquad \nu=w(T_1)-\lambda T_1.
$$
The divided difference is an integral <formal power series>, including its <tangent line> specialization at $T_1=T_2$. The line $w=\lambda t+\nu$ meets the cubic in the two chosen points and a third point $R$. Substitution in the equation shows that its cubic and quadratic coefficients in $t$ are $-A,-B$, where
$$
A=1+a_2\lambda+a_4\lambda^2+a_6\lambda^3,\qquad
B=a_1\lambda+a_2\nu+a_3\lambda^2+2a_4\lambda\nu+3a_6\lambda^2\nu.
$$
Since $\lambda$ has total degree at least two and $\nu$ at least three, $A$ is a unit. The sum of the roots therefore gives
$$
t_R=-T_1-T_2-B/A,\qquad w_R=\lambda t_R+\nu,\qquad
F(T_1,T_2)=-\frac{t_R}{1-a_1t_R-a_3w_R}.
$$
The last expression reflects the third intersection according to the <chord-and-tangent group law>. Both denominators have constant term one. This proves integrality of the universal construction and also provides an explicit coefficient algorithm. Expanding it, the resulting <formal group of an elliptic curve> begins
$$
\boxed{F(T_1,T_2)=T_1+T_2-a_1T_1T_2-a_2(T_1^2T_2+T_1T_2^2)+O((T_1,T_2)^4).}
$$
The integral coefficient recursion and the unit-denominator addition calculation work universally over $\mathbb Z[a_1,a_2,a_3,a_4,a_6]$; thus these coefficients specialize in every <characteristic of a field>. This is the <initial coefficients of the elliptic formal group at infinity> construction, rather than a definition requiring division by the <elliptic-curve discriminant>.
The identity, commutativity and associativity of the <elliptic curve group law> give the defining identities of a one-dimensional commutative <formal group law>:
$$
F(T,0)=T,\qquad F(X,Y)=F(Y,X),\qquad F(F(X,Y),Z)=F(X,F(Y,Z)).
$$
Reflection in the generalized <Weierstrass equation of an elliptic curve> gives its <formal inverse> explicitly:
$$
\iota(T)=-\frac{T}{1-a_1T-a_3w(T)}.
$$
Successive addition gives the multiplication series $[n]_F(T)=nT+O(T^2)$.
In <characteristic zero>, an especially useful description comes from the <invariant differential on an elliptic curve>:
$$
\omega=\frac{dx}{2y+a_1x+a_3}=\bigl(1+a_1t+(a_1^2+a_2)t^2+O(t^3)\bigr)\,dt.
$$
Integrating with zero constant term constructs the <formal logarithm>
$$
L(t)=t+\frac{a_1}{2}t^2+\frac{a_1^2+a_2}{3}t^3+O(t^4).
$$
Translation invariance of the <invariant differential on an elliptic curve> implies $L(F(X,Y))=L(X)+L(Y)$. Thus the <formal logarithm> converts the <formal group law> into the <formal additive group> in characteristic zero. Expanding this identity to degree three also recovers the displayed coefficients of $F$; in positive characteristic the integral addition construction remains valid although the logarithmic denominators may not exist.
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