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Sequences and Series

- The short-short version ?!

What is a sequence?

- A sequence can be thought of as an infinite

list of numbers in order. More formally, as a

function from the set of positive integers to the

set of reals. - Examples
- 1) an 1, 2, 3, . or an n2) an 1,

or an 1/n3) an

or an 1/2n

Find the first 5 terms of the following sequences.

- an n/(n1)
- an
- 2) an (-4)/n!
- an

What can we do with sequences?

- As sequences are functions, we can
- - graph them,
- - we can find their limit at infinity, (if the

limit exists and is a real number we say the

sequence is convergent, otherwise the sequence is

divergent) - - we can decide if they are increasing or

decreasing, - - we can decide if they are bounded

What is a series?

- A series or infinite series is simply the sum

of infinitely many terms - a1 a2 a3 an
- Short hand or

What does it mean to add up infinitely many terms?

- Consider an infinite series
- and let sn a1 a2 an
- be the nth partial sum of the series,
- then we say that
- a) if the sequence s is convergent and the

limit as n approaches infinity exists as a real

number, say s, then the series converges and we

write - a1 a2 an s or
- b) otherwise, the series diverges.

The Geometric Series

- A geometric series is of the form
- a ar ar2 ar3 arn-1
- It converges if r lt 1 and
- its sum is a/(1-r). Otherwise, the geometric

series is divergent.

Examples

- Find the sum of the geometric series
- a) 1 0.4 0.16 0.064
- b) 2 2/3 2/9 2/27 2/81
- c) 0.1111111

Power Series

- A power series is of the form
- c0 c1 x c2 x2 c3 x3
- where the cs are the coefficients and the x

is a variable.

Example

- If we let the constants be all 1 in the power

series, then we get the geometric series - 1 x x2 x3 1/(1-x)
- if x lt 1, which is the radius of

convergence of this power series.

Note

- We can use power series to represent certain

functions, with radius of convergence, which

allows us to both differentiate and integrate

functions using this form term by term. - Question becomes which functions are

representable?

Taylor Series

- If a function f has a power series expansion

at some number a, then it has the form - f(x)
- this is called the Taylor series of the

function f at a (or centered at a).

Maclaurin Series

- The special case of the Taylor series when a

0 is simply of the form - f(x)
- Lets look at some examples!

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