Calculus: Analytic Geometry and Calculus, with Vectors

606 Series

and show that

n-1

(2)

j

o n2 + V >Jonn2 + x2dx = n tan 1

n]o

Use this result to show that

n-1

(3) 2 2=.

n=1 k=o n I k

Remark: This result is of interest in cosmology. Suppose a particular universe

contains an earth at the origin of a plane x, y coordinate system and contains a

star like our sun at each point (x,y) for which x and y are integers not both zero.

The rate (in appropriate units) at which the earth receives radiated energy from

the star at the point (n,k) is then

1

n2 + k2

provided the inverse square law is applicable These hypotheses and (3) imply

that the earth would receive energy at an infinite rate and hence would burn up

instantly. This result implies that either the stars cannot be so uniformly dis-

tributed or that (perhaps because other stars and interstellar material interfere

with transmission of energy) the inverse square law is inapplicable.

14 Supposing that 0 < s < 1, show that

jn' 1 a -

k=1 n1-sl + C. (s),

where 0 < CG(s) <_ 1. Show that

n 1

1 -1 =2--2+C,,

k=1 ""

where 0 < C. <= 1. Check the last result when n = 1 and when n = 4.

15 Prove that, when n >= 2,

n 1 n-1 1

kIk log k (k + 1) log (k + 1)

= log log n - log log 2 + Cn,

where 0 < Cn <= 1/(2 log 2).

(^16) Prove that, when s > 1 and n > 2,
fin'^1 _^111

k 2k(log k)' -l [(log 2)'-' - (log n)"-I +

where 0 5 Cp(s) 5 1/2(log 2)', and that

k k(log k)' - s (^1) 1 (log 2)'-1+ C(s),
where 0 S C(s) 5 1/2(log 2)'.