Advanced High-School Mathematics

276 CHAPTER 5 Series and Differential Equations

(c)

∑∞

n=1

n^2

√

n^7 + 2n

(d)

∑∞

n=0

n^2 + 2n

3 n

(e)

∑∞

n=0

(n+ 1)3n

n!

(f)

∑∞

n=1

2 n

nn

(g)

∑∞

n=1

n!

nn

(h)

∑∞

n=1

(^1

1 +^1 n

)n

2. As we have already seen, the series

∑∞

n=1

1

n^2

, converges. In fact, it is

known that

∑∞

n=1

1

n^2

=

π^2

6

; a sketch of Euler’s proof is given on page

228 of the Haese-Harris textbook.^11 Using this fact, argue that

∑∞

n=0

1

(2n+ 1)^2

=

π^2

8

.

3. Prove thesinusoidal p-series test, namely that

∑∞

n=1

sin

(

2 π

np

)



converges ifp > 1 ,

diverges ifp≤ 1.

(Of course, the 2πcan be replaced by any constant. Exercise 2 on

page 256 is, of course, relevant here!)

4. Recall Euler’sφ-functionφ(see page 63). Determine the behavior
   of the series

∑∞

n=1

1

nφ(n)

.(See Exercise 16d on page 64.)

5. How about

∑∞

n=1

φ(n)

n^2

?

6. Let F 0 , F 1 , F 2 , ... be the terms of the Fibonacci sequence (see
   page 93). Show that

∑∞

n=0

Fn

2 n

converges and compute this sum ex-

plicitly. (Hint: you’ll probably need to work through Exercise 7

on page 106 first.)

(^11) Peter Blythe, Peter Joseph, Paul Urban, David Martin, Robert Haese, and Michael Haese,
Mathematics for the international student; Mathematics HL (Options), Haese and
Harris Publications, 2005, Adelaide, ISBN 1 876543 33 7