Mathematical Methods for Physics and Engineering : A Comprehensive Guide

SERIES AND LIMITS

4.15 Prove that

∑∞

n=2

ln

[

nr+(−1)n

nr

]

is absolutely convergent forr= 2, but only conditionally convergent forr=1.

4.16 An extension to the proof of the integral test (subsection 4.3.2) shows that, iff(x)
is positive, continuous and monotonically decreasing, forx≥1, and the series
f(1) +f(2) +···is convergent, then its sum does not exceedf(1) +L,whereL
is the integral
∫∞

1

f(x)dx.

Use this result to show that the sumζ(p) of the Riemann zeta series

∑

n−p,with

p>1, is not greater thanp/(p−1).

4.17 Demonstrate that rearranging the order of its terms can make a condition-
ally convergent series converge to a different limit by considering the series∑
(−1)n+1n−^1 =ln2=0.693. Rearrange the series as

S=^11 +^13 −^12 +^15 +^17 −^14 +^19 + 111 −^16 + 131 +···

and group each set of three successive terms. Show that the series can then be

written

∑∞

m=1

8 m− 3

2 m(4m−3)(4m−1)

,

which is convergent (by comparison with

∑

n−^2 ) and contains only positive

terms. Evaluate the first of these and hence deduce thatSis not equal to ln 2.

4.18 Illustrate result (iv) of section 4.4, concerning Cauchy products, by considering
thedoublesummation

S=

∑∞

n=1

∑n

r=1

1

r^2 (n+1−r)^3

.

By examining the points in thenr-plane over which the double summation is to

be carried out, show thatScan be written as

S=

∑∞

n=r

∑∞

r=1

1

r^2 (n+1−r)^3

.

Deduce thatS≤3.

4.19 A Fabry–P ́erot interferometer consists of two parallel heavily silvered glass plates;
light enters normally to the plates, and undergoes repeated reflections between
them, with a small transmitted fraction emerging at each reflection. Find the
intensity of the emerging wave,|B|^2 ,where

B=A(1−r)

∑∞

n=0

rneinφ,

withrandφreal.