Calculus: Analytic Geometry and Calculus, with Vectors

12.1 Definitions and basic theorems 589

series is divergent. A little thought about this matter can lead us to the
idea that a series Muk cannot be convergent unless un - 0 as n - x. To
prove that the idea is correct, suppose Euk converges to s. Then sn --- s
and sn_1-). s as n -> x and, since un = sn - sn_1, it follows that u^0
as n -- x. Several ideas can be obtained by investigating the series

(12.137) 2+ 3+4+4+I+- 5

1 _ +

5 5

and its sequence of partial sums. The nth term of the series approaches

0 as n --> -. As we plot the points sl, S2, S3, ss, ' ' , wefind ourselves

hopping to and fro between 0 and 1. The sequence of partial sums is

bounded, but the series is not convergent. The possibility of learning

about convergence of series will be enhanced if we obtain a full apprecia-

tion of the way in which the following fundamental theorem is proved.

Theorem 12.14 If

(12.141) s = ul+u2+u3+

and

(12.142) t = vl + v2 + v3 +

and if a and b are constants, then

(12.143) as + bt = (aul + bv1) + (au2 + bv2) + (aua + bv3) +

To prove this theorem let, for each n = 1, 2, 3,. ,

X. = 211+ Z62+. .. +un, to = v1+V2+... +5n

Rules of arithmetic (or possibly algebra) allow us to multiply by a andb,

respectively, and add the results to obtain

asn + bin = (au1 + bv1) + (au2 +bv2) + ... + (aun + bvn).

Thus asn + bin is the sum of n terms of the series in (12.143),and to

prove (12.143), it is necessary to prove that

(12.144) lim (asn + btn) = as + bt.

n-.-

It is now very easy to see how to finish the proof. The hypothesis

(12.141) means that the first of the formulas

(12.145) urn s,, lim to = t

n-. n-pm

holds, and the hypothesis (12.142) means that the second holds. Finally

(12.145) implies (12.144) and the proof is finished. The above theorem

and the next two lie at the foundation of the theory of series.