Series

DEFINITION: An infinite series is an expression that can be written in the form



The numbers are called the terms of the series.

Consider



DEFINITION: Let be the sequence of partial sums of the series



If the sequence converges to a limit then the series is said to converge to and is called the sum of the series. We denote this by writing



If the sequence of partial sums diverges, then the series is said to diverge. A divergent series has no sum.

Famous Series

1.

2. (Harmonic series)

3.

4.

5.

6.

  

7.

Notation

DEFINITION:



EXAMPLES:

(a) 

(b) 

(c) 

(d) 

(e) 

(f) 

(g) 

(h) 

(i) 

(j) 

EXAMPLE: Write



using summation notation.


EXAMPLE: Write



using summation notation.


EXAMPLE: Write



using summation notation.


EXAMPLE: Write



using summation notation.


EXAMPLE: Write



using summation notation.


EXAMPLE: Write



using summation notation.


EXAMPLE: Write



using summation notation.



Geometric Series

THEOREM: A geometric series



converges if and diverges if If the series converges, then the sum is





EXAMPLE: Find the sum of the geometric series




EXAMPLE: Find the sum of the geometric series




EXAMPLE: Find the sum of the geometric series




EXAMPLE: Find the sum of the geometric series




EXAMPLE: Find the sum of the geometric series




EXAMPLE: Find the sum of the geometric series




PROBLEMS:

(a) Is the series convergent or divergent?


(b) Write as a ratio of two integers.


(c) Find the sum


(d) Find the sum



Telescoping Sums
EXAMPLE: Evaluate




REMARK: In the same way one can prove that



EXAMPLE: Find




EXAMPLE: Show that



for any positive integer



Divergence Tests

THEOREM (The Divergence Test):

(a) If does not exist or if then the series diverges.

(b) If then the series may either converge or diverge.

THEOREM: If the series converges, then

EXAMPLES:

(a) diverges because does not exist.

(b) diverges because

(c) diverges because



(d) diverges because



(e) We have however diverges and converges.

THEOREM:

(a) If and are convergent series, then and are convergent series and the sums of these series are related by



(b) If is a nonzero constant, then the series and both converge or both diverge. In the case of convergence, the sums are related by



(c) Convergence or divergence is unaffected by deleting a finite number of terms from a series; in particular, for any positive integer the series



both converge or both diverge.

EXAMPLE: Find .


EXAMPLE: Find .