166
II - Convergence: Discrete variables
of x4 is obtained by choosing the factor 1 everywhere except in the factors p
and q, which yields a contribution x4/p2q2; one thus has
7I"4/5! = 2: 1/p2q2,
p
The identity
(2: Xi f = 2: X~ + 22: XiXj
i
etc. Of course, Euler did not worry over such justifications; such extraordinary results were enough to make him happy. He might have checked them
numerically, having discovered very effective and ingenious methods for calculating series such that those we are dealing with here.
His argument for "proving" the relation (9) in 1734 was to observe that
if an algebraic equation of degree n
(21.10)
P(X) = ao + alX + ... + anxn = 0
has n distinct roots Xl, ... , X n , real or complex, then the left hand side of
(10) is identical to the polynomial an(x - Xl) ... (X - xn), a perfectly precise
statement, and easy to prove 53 and that Baccalaureat candidates are even
supposed to know for n = 2. It follows that
(for n = 2, the product cia of the roots of the trinomial ax2 + bx + c), thus
that
P(X)
(21.11)
(-I)nao(x - xt) ... (x - Xn)/Xl ... Xn =
ao(l - X/Xl) ... (1 - x/xn)
if ao =I- o. Now "the algebraic equation of infinite degree"
(21.12)
sinx/x = 1 - x 2 /3! + x 4 /5! - ... = 0,
53 For u E IC given, P(y + u) is a polynomial in y whose term independent of y, the
value for y = 0, is P(u). If now P(u) = 0, then P(y+u) is divisible by y, so that
P(x) = (x - u)Q(x) where Q is a polynomial, with dO(Q) = dO(P) - 1. If v =f u
is another root of P, one has Q(v) = 0, whence P(x) = (x - u)(x - v)R(x), etc.
This argument assumes that the n roots of P are distinct, for if not the factors
x - u, x - v, etc. might be repeated ("order of multiplicity" of a root).
II - Convergence: Discrete variables
of x4 is obtained by choosing the factor 1 everywhere except in the factors p
and q, which yields a contribution x4/p2q2; one thus has
7I"4/5! = 2: 1/p2q2,
p
The identity
(2: Xi f = 2: X~ + 22: XiXj
i
etc. Of course, Euler did not worry over such justifications; such extraordinary results were enough to make him happy. He might have checked them
numerically, having discovered very effective and ingenious methods for calculating series such that those we are dealing with here.
His argument for "proving" the relation (9) in 1734 was to observe that
if an algebraic equation of degree n
(21.10)
P(X) = ao + alX + ... + anxn = 0
has n distinct roots Xl, ... , X n , real or complex, then the left hand side of
(10) is identical to the polynomial an(x - Xl) ... (X - xn), a perfectly precise
statement, and easy to prove 53 and that Baccalaureat candidates are even
supposed to know for n = 2. It follows that
(for n = 2, the product cia of the roots of the trinomial ax2 + bx + c), thus
that
P(X)
(21.11)
(-I)nao(x - xt) ... (x - Xn)/Xl ... Xn =
ao(l - X/Xl) ... (1 - x/xn)
if ao =I- o. Now "the algebraic equation of infinite degree"
(21.12)
sinx/x = 1 - x 2 /3! + x 4 /5! - ... = 0,
53 For u E IC given, P(y + u) is a polynomial in y whose term independent of y, the
value for y = 0, is P(u). If now P(u) = 0, then P(y+u) is divisible by y, so that
P(x) = (x - u)Q(x) where Q is a polynomial, with dO(Q) = dO(P) - 1. If v =f u
is another root of P, one has Q(v) = 0, whence P(x) = (x - u)(x - v)R(x), etc.
This argument assumes that the n roots of P are distinct, for if not the factors
x - u, x - v, etc. might be repeated ("order of multiplicity" of a root).