Series

I.  Quiz

II.  Homework

III.  Definition of a Series

Let a_n be a sequence then we define the n^th partial sum of a_n as

s_n = a₁ + a₂ + ... + a_n

In other words, we define s_n by adding up the first n terms of a_n    

We define the series as the limit of  the s_n that is

S =  sum from i = 1 to infinity of  a_n   =  a₁ + a₂ + a₃ + ...

If the limit exists then we say that the series converges.  Otherwise, we say that the series diverges.  

Example 

consider a_n  = 1/(n² + 2n + 1) - 1/n²

Evaluate S = sum from n = 1 to infinity of  1/(n² + 2n + 1) - 1/n²

Solution

We write out the first four terms: (1/4 - 1/1) + (1/9 - 1/4) + (1/16 - 1/9) + (1/25 - 1/16) + ...

= -1/1 + 1/4 - 1/4 + 1/9 - 1/9 + 1/16 - 1/16 + 1/25 - ... = -1

Such a series is called a telescoping series.

IV.  Geometric Series

We define a geometric series to be a series of the form

sum arⁿ

For example:

3/2 + 3/4 + 3/8 + ...

Theorem:  For 0 < |r| < 1  we have

sum arⁿ = a/(1 - r)

and for |r| > 1 the series diverges.

Proof:

Let

s = a + ar + ar² + ar³ + ar⁴ + ...  Then

rs =   ar² + ar³ + ar⁴ + ...    

subtracting the second equation from the first we get

rs - s = a or

s(1 - r) = a,

s = a/(1 - r)

V.  The Limit Test

Theorem:  If sum a_n converges then the limit as n -> infinity of a_n = 0 

Note:  The contrapositive says that if the limit is nonzero, then the series does not converge.

Caution:  if the limit goes to zero then the series still may diverge.

Examples

A)  sum (n + 3)/(n + 2)  diverges by the limit test since the limit is 1 not 0.

B)  sum 1/n does not converge even though the limit goes to 0.

VI.  The Harmonic Series

Theorem:  the series with terms 1/n diverges

Proof:  we write

1/1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + 1/9 + 1/10 + 1/11 + 1/12 + 1/13 + 1/14 + 1/15 + 1/16 + 1/17 + ...

< 1/1 + 1/2 + 1/4 + 1/4 + 1/8 + /18 + 1/8 + 1/8 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/16 + 1/32 + ...

= 1/2 + 1/2 + 1/2 + ...

which diverges.  Hence the harmonic series diverges.