Calculus: Analytic Geometry and Calculus, with Vectors

12.6 Euler-Maclaurin summation formulas 651

in which the star on the Z means that the term for which is = 1 is omitted from
the series. More advanced mathematics contains theorems whichallo« us to
take termwise limits, as x approaches zero through positive values, of themem-
bers of (7). This, the fact that

(8) lim B,(x) = B (0) -

B.

x-.0 n!

when n 0 1, and the fact that B. = 0 when n is odd and is 0 1, give the formula

t el + B2 Bq B6 B2k

(9) 2 7 °=1 Bo + 2 t2 + 4 t4 +6 t6 + ... =ko (2k)1t2k

which is valid when 0 < Itl < 2x. Putting t = 2z in (9) gives the formula

e= + e 22kB2k yk

(10) zex - e = ko (2k)!z

which is valid when 0 < IzI < ir. The theory of functions of a complex variable
provides reasons why (10) is valid when z is a complex number for which 0 <
IzI < ir, and we can put z = iO to obtain

(11) i8

es0 + e-,e 22kB2k

2k

e'B - e-'B ko (2k)! (i0)

when 0 < 101 < a. Since ilk = (j2)k = (-1)k, use of the Euler formulas gi-'en
in Problem 20 of Problems 12.49 enables us to put (11) in the form

(12) 0 cot 0 = I (-1)k (ZB2i 02k

k=0

when 0 < IBI < 2r. Having established (12), we can show that

(13) 0 tan 0 = 6 cot 0 - 20 cot 26

when 0 < 101 < a/2 and use (12) to obtain our final formula

(14) tan 0 = i (-1)k-, 22k(22k (2k)!1)B2k a2k-i

A;=1

which is valid when 101 < it/2. The above formulas and modifications of them
appear in quite elementary tables, but we must always be prepared to observe
that some brief treatments of Bernoulli numbers use Bk to denote the number
B2k(0)/(2k)! which we have called B2k.