is convergent for all values of x, except only when x is the

square of an integer.

CHAPTER XXII.

THE BINOMIAL THEOREM. ANY INDEX.

278. It was proved in Chapter xx. that, when n is any positive integer,

We now proceed to prove that the above formula is true for all values of n, provided that the series on the right is convergent.

When n is a positive integer the above series stops, as we have already seen, at the (n+1)th term; but when n is not a positive integer the series is endless, for no one of the factors n, n − 1, n − 2, &c. can in this case be zero. It should be noticed that the general term of the n (n − 1) (n − 2)... (n − r + 1)

binomial series, namely

cannot be written in the shortened form

r

n is a positive integer; we may however employ the notation of Art. 237, and write the series in the form

279. Proof of the Binomial Theorem. Represent,

for shortness, any series of the form 1 +

Mr

m

x2 +......,

f(m)=1+

1

[1

12

Now the coefficient of a" in the product ƒ (m) × ƒ (n) is

And, by Vandermonde's Theorem [Art. 245 or 257],

this coefficient is equal to

of x in f (m + n).

(m+n), which is the coefficient

r

Thus the coefficient of any power of x in ƒ (m + n) is equal to the coefficient of the same power of x in the product f (m) × ƒ (n); also the series ƒ (m), f (n) and f(m+n) are convergent, for all values of m and n, when a is numerically less than unity [Art. 274]. It therefore follows from Art. 276 that

f (m) x f (n) = f (m+n)........(a),

for all values of m and n, provided that x is numerically less than unity.

Now it is obvious that ƒ (0) = 1, and that ƒ (1) = (1 + x); we also know that if r be a positive integer ƒ (r) = (1 + x)”.

Hence, by continued application of (a), we have

ƒ (m) × ƒ (n) × ƒ (p) ×...= ƒ (m + n) × ƒ (p) ×...

integers; then taking s factors, we have

{ƒ(6)}* = ƒ (¦ × 8 ) = ƒ (r).

But, since r is a positive integer, ƒ (r) is (1 + x)" ;

This proves the Binomial Theorem for a positive fractional exponent: the theorem is therefore true for any positive index.

And, assuming that the binomial theorem is true for any positive index, it can be proved to be true also for any negative index. For, from (a),

f (−n)xf (n)=f(-n+n)=f(0).

Hence, as ƒ (0) = 1, we have

Hence (1+x) = f (-n), which proves the theorem for any negative index.

280. Euler's Proof. Euler's proof of the Binomial Theorem is as follows.