Proof:

**Case 1:**- Suppose is real for all
, and that
converges. Then

for all , so by the comparison test, converges. Then , being the difference of two convergent sequences, is convergent; i.e., converges. **Case 2:**- Suppose is an arbitrary absolutely convergent complex series. We
know that for all
,

and

so by the comparison test, and are convergent, and by Case 1, and are convergent. It follows that is convergent.

I claim is absolutely convergent (and hence convergent). We have

We have

so by the ratio test, converges. Hence is absolutely convergent, and hence it is convergent. Clearly , so converges for all . In the exercises you will show that is also convergent for all .

Motivated by the results of section 10.3, we make the following definitions:

For all , , let

Then

I would now like to be able to say that for all ,

i.e., I would like to have a theorem that says

However, the next example shows that this hoped for theorem is not true.

and

so

Now . So for all , and thus . Eventually we will show that and , but it will require some work.

- a) For what complex numbers does converge?
- b) For what complex numbers does converge?

Oresme c. 1360 [31, p437]. However, many 17th and 18th century mathematicians believed that (in our terminology) every null sequence is summable. Jacob Bernoulli rediscovered Oresme's result in 1687, and reported that it contradicted his earlier belief that an infinite series whose last term vanishes must be finite[31, p 437]. As late as 1770, Lagrange said that a series represents a number if its th term approaches [31, p 464].

The ratio test was stated by Jean D'Alembert in 1768, and by Edward Waring in 1776[31, p 465]. D'Alembert knew that the ratio test guaranteed absolute convergence.

The alternating series test appears in a letter from Leibniz to Jacob Bernoulli written in 1713[31, p461].

The series (11.35) for is called *Mercator's formula* after Nicolaus Mercator
who published it in 1668. It was discovered earlier by Newton in 1664
when he was an
undergraduate at Cambridge. After Newton read Mercator's book, he quickly wrote down
some of his own ideas (which were much more general than Mercator's) and allowed his
notes to be circulated, but not published. Newton used the logarithm formula to
calculate to 68 decimals (of which the 28th and 43rd were wrong), but a
few years later, he redid the calculation and corrected the errors.

See [22, chapter 2] for a discussion of Newton's work on series.

The series representation for (11.37) is called *Gregory's
formula* after John Gregory (1638-1675) or *Leibniz's formula* after Gottfried
Leibniz (1646-1716). However, it was known to sixteenth century Indian
mathematicians who credited it to Madhava (c. 1340-1425). The Indian version was

(See[30, p292].)