1550251515-Classical_Complex_Analysis__Gonzalez_

Sequences, Series, and Special Functions 623

The formula

I z

r( )^1

. n.n
z = 1m
n--+oo z(z + 1) .. · (z + n)
(8.20-19)

was known to Euler and Gauss and it is called .the Euler (or Gauss) formula.

Clearly, the points z = O, -1, -2, ... , -n, ... are simple zeros of l/r(z),

so r(z) has simple poles at the origin and at the negative integers.

(12) The expansion (8.3-11) gives for z = x, with lxl < 1,

ln(l + x) = x -1/ 2 x^2 + 1/ 3 x^3 -^1 / 4 x^4 + · · ·

and

Hence

lxl < 1

and letting x = l/(2n+ 1) (n = 1,2,3, ... ), we have

n+l [ 1 1 1 1 1 ]

ln -n-=^2 2n + 1 + 3 (2n + 1) 3 + 5 (2n + 1)^5 + ...

which implies that

2

• < ln (1 + .!.)
  2n+ 1 n

< 2n~l + ~ (2n~l)2 [2n~l + (2n~l)^3 + .. ·]
or

--<ln^2 ( l+-1) <--+-------^2 2 1
2n+l n 2n+1 3 4n(n+l)(2n+l)
or

1 < ( n + ~) ln ( 1 + ~) < 1 + 12 n(! + 1 )

and we may write

(8.20-20)

where

1 1 (1 1 )

O<an < 12n(n+l) = 12 n - n+l

(8.20-21)