Sequences and Series

 

 

What's the Difference?

A sequence is any list of numbers that can be finite or infinite. We are interested in finding rules that can be applied to them,

e.g. 1,3,5,7,9 can be written as:

A series is the sum of a number of terms in the sequence,

e.g. if then
This can also be written as:

 

Convergent and Divergent Sequences

A sequence converges if the terms tend to a particular number as n gets larger. This number is called the limit.

e.g.(the sequence converges to 2, so the limit is 2)

(convergent to 0)

The rules for the limits of polynomial fractions such as the one above are:

• If the bottom is of higher order, then the sequence is convergent to 0. E.g.
  
• If the top is of higher order (it contains a higher power), then the sequence is divergent to +¥ . E.g.
  
• If top and bottom are of the same order, then the sequence converges to the ratio of the coefficients of the highest power. (see above)

 

Periodic Sequences

These are sequences of regularly repeating values. You will usually be asked for the period, i.e. the number of terms before the sequence starts repeating itself,

E.g. 1,2,3,1,2,3,1,2,3 has period 3

 1,2,3,2,1,2,3,2,1 has period 4

 sin x has period 2p

 

 

Oscillating Sequences 

Oscillating sequences are those where the values of the terms neither converge nor diverge, they move around. Common types are those which oscillate in sign,

E.g. begins with -1, 2,-3, 4,-5 etc.

 Begins with 1,-2, 4,-8,16,-32 etc.

 Note that periodic sequences are also oscillating, since they neither converge nor diverge.

 

 

Arithmetic Series

The definition is in the formula book:

Note that a is the 1^st term, d is the common difference.

 So, for example, in the progression 5, 2,-1,-4,-7,-10, a=5 and d=- 3

 All arithmetic series are divergent. (i.e. the sum to infinity always adds up to +or - ¥ )

The formula for the sum to n terms is in the formula book:

 

 

 

Geometric Series

Again, the definition is given in the formula book:

Note that a is the 1^st term and r is the common ratio.

 So the progression 6,-18,54,-162 has a=6 and r=- 3

 You need to remember that if r is between - 1 and +1, both the series and the sequence converge . For a series to converge means that the sum to infinity adds up to a particular number, so obviously the sequence must converge to 0.

In this case, the formula is in the formula book:

 

If r > 1 the sequence and series will diverge.

 If r < - 1 the sequence will oscillate infinitely.

 The general formula for the sum to n terms is also given in the formula book: