Calculus: Analytic Geometry and Calculus, with Vectors

650

x = t/2 then gives the formula

(21) lim Zn (k) = 1 f A

n-a=

2 4 2 n

Series

The right side of (21) can be evaluated sith the aid of tables giving values of

1

we'212dt

Jo

for various values of co.

6 Our educations are not quite complete until we have seen the formula by

which the power series expansion of tan x is expressed in terms of Bernoulli

numbers Letting Bo(x), B1(x), be the Bernoulli functions, we start by

deriving the formula

(1)

ze-

of - 1 _n=u) Bn(x)tn,

which holds when 0 < x < 1 and 0 < Itl < 2r. To simplify matters, we can

suppose at first that 0 < jtj < 1. When y(x) denotes the right side of (1), we

can differentiate termwise to obtain

(2) Y'(x) _ Bn(x)t" = 4 B;,+1(x)t"+'

n=1 n=0

Therefore,

(3)

m m

= jj Bn(x)t"+' = t jj B"(x)tn = ty(x)

n-0 n-0

d [e z'y(x)t = e xt[y'(x) - ty(x)J = 0,

and it follows that for each t there is a constant c(t) for which

(4) a zty(t) = c(t) or y(t) = c(t)ezt.

Therefore,

(5) c(t)ez' = I Bn(x)tn

n =O

Integrating (5) over the interval 0 5 x < 1 and using (12.611) and (12.613) give

the first and then the second of the formulas

(6) c(t)

et

t

1

= 1, c(t) =

t

1

Substituting in (5) then gives (1). Since B,(x) = x - I when 0 < x < 1, we

can subtract (x - $)t from both sides of (1) to obtain the formula

(7) 1 el + 2ex' - 1 - 2x(et - 1)= j* B"(x)tn

e'-1 n-0