SERIES AND MATHEMATICAL INDUCTION 15

given series, but merely enables us sometimes to prove the
correctness of a formula that is said or supposed to represent
that nth term.

Induction will sometimes help us to find a formula for the
nth term and then mathematical induction may furnish the
proof.

MULTIPLICATION OF THE TERMS OF A SERIES

39. Def. Another important method of obtaining a second
series from a given series a1, s, as, -+ - is to multiply the terms
instead of adding them. We will use the symbol P, to denote
the product of the first # terms of the given series.

Thus B
Pg =.ay X @
P3=.a1><az><a3

Po=a1 X @ X s X »+» X au,if > 1
Also Py = P; X as
PaszXaz.

Pn¥Pn_1Xan,ifn>1
Pn+1=PnXan+1

The new series Py, Ps, P3, ---, P,, +-+ may be called the
series of products of the original series.

k=n
40. By analogy with § 28 we may represent P, by II (az),

k=1
a symbol intended to indicate that in the expression in the
parenthesis we first put £ = 1, then 2 = 2, then k£ = 3, and
so on up to & = #n, and multlply the results obtaind.

This symbol is sometimes shortend to II (ax) or to II (ax)

The symbol II, pi, is the Greek letter corresponding to the
Roman, or English, letter P.