Advanced High-School Mathematics

296 CHAPTER 5 Series and Differential Equations

5. Consider the function defined by settingf(x) = ln(1 + sinx).

(a) Determine the Maclaurin series expansion for f(x) through

thex^4 term.

(b) Using (a) determine the Maclaurin series expansion for the

functiong(x) = ln(1−sinx).

(c) Using (a) and (b), determine the Maclaurin series expansion

for ln secx.

6. Consider the following integral

I =

∫ 1

0

(∫ 1

0

dy

1 −xy

)

dx.

(a) Using integration by parts, show that the internal integral is

equal to

−ln(1−x)

x

.

(b) Determine the Maclaurin series expansion for this.

(c) Use a term-by-term integration to show that

I =

∑∞

n=1

1

n^2

,

(which, as mentioned on page 276 is =π

2

6 .See the footnote.

(^20) )

7. Here’s a self-contained proof that

∑∞

n=1

=

π^2

6

.(See the footnote.^21 )

Step 1. To show that for any positive integerm,

∑m

k=1

cot^2

kπ

2 m+ 1

=

m(2m−1)

3

.

To complete step 1, carry out the following arguments.

(^20) Using the variable changeu=^12 (y+x), v=^12 (y−x), one can show in the above “double
integral” is equal toπ 62 , giving an alternative proof to that alluded in HH, Exercise 15, page 228.
Showing that the double integral has the correct value requires some work!
(^21) This is distilled from I. Papadimitriou,A Simple Proof of the Formula ∑∞
k=1
k^12 = π 62 .Amer.
Math. Monthly 80 , 424–425, 1973.