Calculus: Analytic Geometry and Calculus, with Vectors

594 Series

by showing that if s" is the sum of n (that is, the first n) terms of the series, then

s"=(1- +( +(

-4)-I-

+`n n I d 1

Note that the middle sum is a telescopic sum.

10 With the aid of a comparison of the two series

42+ 1 +6z+ I +8z+ ... 34+ 5+56+67+78+ ...

prove the first of the formulas

z

1 < l + ++s=fi,

_7r

k1k° k1k2^6

The second formula is a simple consequence of basic theory of series known as

Fourier series. The result we have obtained is significant because 1r2 is about

z

10 and 6 is about 4 or y6-0g.

(^11) The first of the series in
(^1111) q-13+3.5+5.7+79+... 1 1 1
and 1=1-22 i -!-.3 13-4 4.5 -
converges because it is dominated by the second and the second is convergent.
Show that 0 < q < 1 and, if possible, find q.
12 Prove that if ak > 0 and 2;ak < oo, then Eak < 00.
13 This is a preliminary skirmish with the harmonic series

(1) 1 + -F s -f-.+ + -F fi -1- g -I-$ ....

This series is divergent because its terms and partial sums are greater than or

equal to those of the divergent series

(2) V1

of positive terms. Let H(1), H(2), H(3), ... denote the partial sums of the

harmonic series and, with the aid of (2), show that H(2°) = 1, H(21)

H(22) > $, H(21) > "ff, and, in general,

H(2") > (n + 2)/2.

14 Note that the series in

s-123 + 34+345+456+567+...

is dominated by the series in