**Definition**. Given an integer the -th *harmonic number* is defined by

**Remark**.** **i)

ii)

iii) where is the Euler’s constant (see Problem 2 in this post!).

iv)

v) for This easily follows from estimating the area under the curve using rectangles.

vi) for This is easy to see because

**Problem 1**. (Maclaurin series of and ). Show that

i) for

ii) for

**Solution**. i) By part vi) of the above remark, for and so integrating gives

for and the result follows. The only thing left is to show that is convergent at and that follows from the alternating test (see Exercise 2 for a hint!).

ii) For we have

The only thing left to prove is that is convergent at and that follows from the alternating test (see Exercise 3).

**Problem 2**. The polylogarithm function was introduced in this post. Show that

i) for

ii) for

**Solution**. i) By Problem 1, i) and the first part of the above remark, we have

and the result follows.

ii) By Problem 1, ii), we have

and the result follows from i).

**Example**. Show that

i)

ii)

iii)

**Solution**. For i), put in Problem 2, i), and for ii), put in Problem 2, ii). Notice that by Problem 1, i), in this post,

To prove iii), put in Problem 2, ii), to get

by Problem 1 in this post.

**Exercise 1**. Given positive integers show that

*Hint*.

**Exercise 2**. Consider the sequence Show that is decreasing and

*Hint*. It’s easy to directly show that and by the part iii) or vi) of the above remark.

**Exercise 3**. Consider the sequence Show that is decreasing and