The Binomial Theorem (see “Double Feature on Christmas Day“, “Prelude to Taylor’s theorem“) states:

Provide and are suitably restricted, there is an *Extended* Binomial Theorem. Namely,

where

Although Issac Newton is generally credited with the Extended Binomial Theorem, he only *derived* the germane formula for any rational exponent (i.e., ).

We offer a complete proof as follows:

Let

Differentiate (1) with respect to yields

We have

That is,

From

we see that is a solution of initial-value problem

By we also have

Express (3) as

and then differentiate it with respect to

gives us

i.e.,

Since

we see that is also a solution of initial-value problem (2).

Hence, by the Uniqueness theorem (see Coddington: An Introduction to Ordinary Differential Equations, p. 105),

And so,

Consequently, for

Multiply throughout, we obtain

i.e.,

Prove is convergent:

*Proof*

Prove

*Proof*

that is,

Therefore,

Prove

*Proof*

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