SERIES AND MATHEMATICAL INDUCTION 9

SUMMATION
27. Def. When a series of numbers ay, @3, @3, *++*, Gn, ***
is given, it is sometimes useful to obtain from it a second
series Sy, Ss, S3, + -, S,, ¢+ as follows:
Let S; = oy
Se = a1 + a

Ss%al—{—az'f‘aa

Sn¥a1+02+a3+"'+an,ifn>1(1)

Evidently Se = S1 + a2
S3 =S+ as
Sy = Ss + a4

RSt
Sn+1 = Sn + Ant1

The new series Sy, Ss, S3, -+, Sy, -+ - is called the series of
sums of the original series. It is said to be obtaind from the
original series by summation. Its #th term is the sum of the
first » terms of the original series. This explains why the
letter .S, first letter of ““sum,” is used in the symbol S,.

28. The sum of the » numbers a1, as, as, -+ -, @, is sometimes
represented by the symbol

k=n
2 (ax)
k=1
k=n
Here the symbol )| indicates that in the expression a; we

k=1
are first to put £ = 1, then & = 2, then £ = 3, and so on up
to £k = n, and add the results obtaind.

(1) Most text books on algebra speak of a1 + a2 +as+ +++ + a, as a
series. Rather this is the sum of the first # terms of a series. The series
is ai Gz, @3, ***, Qn.