Calculus: Analytic Geometry and Calculus, with Vectors

12.4 Power series 631

Making wholesale use of the rule of Problem 8 for multiplication of series, show
that

(5)

(^111) °° * 1
1-^1 1-

1 1- 1

n_1n

Pt P2 Pk

where the star on the sigma means that some of the terms for which is >
are omitted from the series. Prove the formula

(6) net * 1na< 1 =S()s

n=1 n=1 n=1

71a

and use it to prove that

(7)

Pk

lim 1 1 1 1 1 1 ... 1 1= (s)

"°1--1--1-- 1--

Ps P2 Pa Pk

Use (5) to obtain the inequality

(8) Pi pi pk >

A P82-1 pk-1- Ca n.n=1

and show that taking the limit as s -> 1 + gives the inequality

(9) p1 P2 Pk y + log

• 1 P2 - 1 pk - 1 = S Pk
  where y is the Euler constant. Show that neglecting the 'y and taking logarithms
  gives

(10) 1 log(1 + 1 > log log p,,.

k1 Pk -

Show that x > log (1 + x) when x > 0 and hence that

jn' 1 > log log

4 pk-1 Pri

and

(12) >1 log logpn.

k1Pk

Show finally that

(13) 1 + 2 + 3 + ... °o.

26 The theory of functions of a complex variable provides an elegant proof

of the fact that if

(1) f(x) = co + ci(x - a) + c2(x - a)2 + cs(x - a)2 +

when IxI < R, then

(2) f'(x) = c1 + 2c2(x - a) + 3cs(x - a)2 +.