Calculus: Analytic Geometry and Calculus, with Vectors

(^616) Series
6 Supposing that x = r/4, give one or more reasons why the series
sin x sin 2x sin 3x sin 4x

1 2 3 4 +

is not an alternating series to which Theorem 12.31 applies.

7 The function 4k being defined by (12.32), prove the formulas (12.33).

8 It is not expected that the theory of Fourier series and its formulas have

been learned, but it is expected that we can start solving problems when suitable

formulas and instructions are given. Write the formulas (12.32) for the case in

which L = I and show that the formula (12.36) for the Fourier coefficients

becomes

ak = o

i

f (x) sin krx dx.

Letting f (x) be the Bernoulli function B, (x) so that f (x) = x - - when 0 < x < 1,

show that

ak =

oi

(x - ) sin krx dx.

Show that integration by parts gives

1l -1 I

j

1

ak = [(x-2/ kr cos krxJO + f0 cos krx dx

-'/21+coskr - Nr2 1+(-1)k

kr 2 - k,r 2

so that ak = 0 when K is odd and ak = - -/kr when k is even. Observe

that the conditions in the sentence following (12.37) are satisfied and hence that

(12.37) must be valid. Then show that substituting in (12.37) gives the first

of the formulas (12.384).

(^9) Another particularly important example involves the square sine function
(or square wave function) defined by
Sin x = sgn sin x.
The graph of this function is shown in Figure 12.391. To find the trigonometric
Y
-z
Figure 12.391
Fourier series of the odd function Sin (rx/L) which has period 2L, use the ortho-
normal set (12.32) and, after observing that Sin (rx/L) = 1 when 0 < x < L,
calculate the Fourier coefficients of Sin (irx/L) from the formula
ak =f oL sin Lx dx.