4. Series of Numbers
4.1. Series and Convergence
So far we have learned about sequences of numbers. Now we will investigate what may happen when we add all terms of a sequence together to form what will be called an infinite series.The old Greeks already wondered about this, and actually did not have the tools to quite understand it This is illustrated by the old tale of Achilles and the Tortoise.
Example 4.1.1: Zeno's Paradox (Achilles and the Tortoise)  
Achilles, a fast runner, was asked to race against a tortoise. Achilles can run 10 meters per second, the tortoise only 5 meter per second. The track is 100 meters long. Achilles, being a fair sportsman, gives the tortoise 10 meter advantage. Who will win ? 
 Both start running, with the tortoise being 10 meters ahead.
 After one second, Achilles has reached the spot where the tortoise started. The tortoise, in turn, has run 5 meters.
 Achilles runs again and reaches the spot the tortoise has just been. The tortoise, in turn, has run 2.5 meters.
 Achilles runs again to the spot where the tortoise has just been. The tortoise, in turn, has run another 1.25 meters ahead.
Obviously, this is not true, but where is the mistake ?
Now let's return to mathematics. Before we can deal with any new objects, we need to define them:
Definition 4.1.2: Series, Partial Sums, and Convergence  
Let { a _{n} } be an infinite sequence.

Examples 4.1.3:  

Definition 4.1.4: Absolute and Conditional Convergence  
A series
converges absolutely if the sum of the absolute values
converges.
A series converges conditionally, if it converges, but not absolutely. 
Examples 4.1.5:  

Theorem 4.1.6: Absolute Convergence and Rearrangement  
Let
be an absolutely convergent series. Then any rearrangement of
terms in that series results in a new series that is also
absolutely convergent to the same limit.
Let be a conditionally convergent series. Then, for any real number c there is a rearrangement of the series such that the new resulting series will converge to c. 
Theorem 4.1.8: Algebra on Series  
Let
and
be two absolutely convergent series. Then:

Theorem 4.1.9: Cauchy Criteria for Series  
The series
converges if and only if for every
> 0 there is a positive
integer N such that if m > n > N then
  < 
Our final story is frequently called the "Leaning Tower of Lire". While the introductory story about Achilles and the Tortoise introduces an apparent paradox which we were able to resolve using a convergent (geometric) series, this story uses the properties of a divergent (harmonic) series to shed light on an unbelievable but true situation.
Example 4.1.10: The Leaning Tower of Lire  
Jillian, a diligent but overworked student, fell asleep in the library and
got locked in for the night. When she awoke, the room was dimly lit and she
was alone. To pass the time (and to annoy the librarian in the morning) she
decided to stack books on a table so that they would overhang the edge
of the table.
Assuming she has an unlimited supply of books, all of equal width 2 and weight 1 (say), what is the biggest overhang she can produce? To make it more interesting, let's say she can use only one book at each level. 
But our storytime is over  the next section introduces convenient tests to determine quickly and efficiently whether a series converges or diverges.