618 
SERIES. 
[796 
of the series. For instance, for the series of odd numbers l + 3 + 5+ 7+ ...,we have 1 = 1, 
1 + 3 = 4, 1 + 3 + 5 = 9, &c. These results at once suggest the law, l+3 + 5 + ...+(2w—1)=m 2 , 
which is in fact the true expression for the sum of n terms of the series ; and this 
general expression, once obtained, can afterwards be verified. 
3. We have here the theory of finite series: the general problem is, u n being 
a given function of the positive integer n, to determine as a function of n the sum 
u 0 + Uj + w 2 + ... + u n , or, in order to have n instead of n +1 terms, say the sum 
Uo + Ui + u 2 + ... + i • 
Simple cases are the three which follow. 
(i) The arithmetic series, 
a + (ci + 6) + (a + 26) + ... + (a + n — 1) 6 j 
writing here the terms in the reverse order, it at once appears that twice the sum 
is = 2a + n —16 taken n times: that is, the sum =nct + ^n(n— 1)6. In particular, we 
have an expression for the sum of the natural numbers 
1 + 2 + 3 + ... + n = \n (n + 1), 
and an expression for the sum of the odd numbers 
1 + 3 + 5 + ... + (2n — 1) = n\ 
(ii) The geometric series, 
a + ar 4- ar 2 + ... + ar n ~ x ; 
here the difference between the sum and r times the sum is at once seen to be 
. 1 — v n 
= a — ar n , and the sum is thus = a ——— ; in particular, the sum of the series 
1 
1 + r 4- r 2 + ... + r n ~ Y = — . 
1 — r 
(iii) But the harmonic series, 
1 1 1 1 
a^"a + 6"^a+26^" "‘ + a + (n — 1)6’ 
or say y + ^ + 3 ... + -, does not admit of summation ; there is no algebraical function of 
n which is equal to the sum of the series. 
4. If the general term be a given function n n , and we can find v n a function of n 
such that v n+1 — v n = u n , then we have u 0 = v 1 — v 0 , iq = v 2 — v 1} u 2 = v s — v 2 ,..., u n = v n+1 — v n ; 
and hence u 0 + u 1 + u 2 + ... + u n = v n+1 — v 0 ,—an expression for the required sum. This 
is in fact an application of the Calculus of Finite Differences. In the notation of 
this calculus v n+1 — v n is written Av n ; and the general inverse problem, or problem of 
integration, is from the equation of differences Av n = u n (where u n is a given function 
of n) to find v n . The general solution contains an arbitrary constant, v n = V n + G ; but 
this disappears in the difference v n+1 — v 0 . As an example consider the series 
w 0 + iq + ... + u n = 0 + 1 + 3+ ... + 2~m(m + 1);