10 SPECIAL TOPICS IN THEORETICAL ARITHMETIC

Example. For the series in which ¢, = n(n — 1)
k=1

2 (@) =a1=0

k=1
k=2

S(a) =a1+as=0+4+2=2

k=1

k=3
() =a1+ar+az=0+2+6=8
k=1

Thus for the series a4, a2, as, *++, @n, ***
k=n
Sn = Z (ak)
k=1

Rie
Sometimes the symbol Z is abbreviated to Z or to Z
k=r
The symbol ), sigma, is the Greek letter corresponding to
the Roman, or English, letter .S.

It may be noted that the result indicated by the symbol
k=n

2 (aw), tho exprest in terms of %, is independent of k.
k=1

29. Sometimes a formula can be obtaind for S, in terms
of n, exprest by means of well known operations. We will
illustrate by a few examples.

30. First consider the series of natural numbers
19 21 374! 5! 6) Sy
whose nth term is #.

Here b |
=12 =3
S3=14+2+4+3=6
Si=14+24+3+4=10