1540470959-Boundary_Value_Problems_and_Partial_Differential_Equations__Powers

1.5 Operations on Fourier Series 89

in whichαis a positive parameter. For this function we havea 0 =0,an=e−nα,
bn=0. By the integral test, the series

∑

nke−nαconverges for anyk. Therefore
fhas derivatives of all orders. The Fourier series off′andf′′are

f′(x)=

∑∞

n= 1

−ne−nαsin(nx),

f′′(x)=

∑∞

n= 1

−n^2 e−nαcos(nx). 

EXERCISES

1.Evaluate the sum of the series

∑∞

n= 11 /n^2 by performing the integration

indicated in Eq. (5).

2.Sketch the graphs of the periodic extension of the function

f(x)=π− 2 x, 0 <x< 2 π,

and of its derivativef′(x)and of

F(x)=

∫x

0

f(t)dt.

3.Suppose that a function has the formulaf(x)=x,0<x<π. What is its

derivative? Can the Fourier sine series offbe differentiated term by term?

What about the cosine series?

4.Verify Eqs. (6) and (7) by integration.

5.Suppose that a functionf(x)is continuous and sectionally smooth in the

interval 0<x<a. What additional conditions mustf(x)satisfy in order

to guarantee that its sine series can be differentiated term by term? the

cosine series?

6.Is the derivative of a periodic function periodic? Is the integral of a peri-

odic function periodic?

7.It is known that the equality

ln

(∣∣

∣∣2cos

(

x

2

)∣∣

∣∣

)

=

∑∞

n= 1

(− 1 )n+^1

n cos(nx)

is valid except whenxis an odd multiple ofπ. Can the Fourier series be

differentiated term by term?