Summation Notation

Summation notation: a way to show the operation of adding a series of values related by an algebraic expression or formula.

The symbol for summation is the Greek letter Sigma, .

#

#n

formulaLower limit

Upper limit

Variable

Algebraic expression

The 20th partial sum of the arithmetic sequence: 2, 7, 12, 17,… would be represented as

Explicit formula of the sequence

You can still use the partial sum formulas to find the answer to these, but you have to make sure you recognize if the expression represents an arithmetic or geometric sequence.

1) 2)

For each sequence below:a)Determine if it is arithmetic or geometric.b)Identify the common difference or ratioc)Write each in summation notation for the 16th partial sum of each sequence.

1) 2)

a) Geometricb) r = 5c)

a) Arithmeticb) d = 8c)

Infinite Sequences & Series

Convergent and Divergent Sequences

Some infinite sequences never approach a unique number and continue forever. These sequences

are called a divergent sequences

Geometric sequence

2, 4, 8, 16, …

Arithmetic sequence

5, 8, 11, 14, …

What conclusion can you make about the type of

sequences that are convergent?

Convergent and Divergent SequencesSome infinite sequences approach a unique

number. (Consider this an asymptote). When the infinite sequence approaches a unique number,

the sequence is a convergent sequence

Example: The geometric sequence 8, 2, ½, , …

What value do the terms

appear to be getting closer

to as the sequence

continues?

If given a sequence, how can you tell if it is a convergent or divergent

sequence?

•Make a list of the first 4 – 5 terms of the sequence and observe the pattern.•Create a graph of the sequence and observe the trend of the points.Determine if each sequence is convergent or divergent.

1) an = -3n+ 12 2) an = 4 · (½)n-1 3) an = 2 · (-3)n-19, 6, 3, 0, -3, …Divergent

4, 2, 1, ½, ¼, …Convergent

2, -6, 18, -54, 162, …

Divergent

An infinite sum means that you add ALL the terms of infinite sequence.

The only type of infinite sequences that can be added are convergent sequences.

Formula for the infinite sum

of a sequence

Remember this is only used on geometric sequences with fractional value for r. -1 < r < 1