Title: Calculus 9.1
19.1 Power Series
2e
0
2
3What You Will Learn
- All continuous functions can be represented as a
polynomial - Polynomials are easy to integrate and
differentiate - Calculators use polynomials to calculate trig
functions, logarithmic functions etc. - Downfall of polynomial equivalent functions is
that they have an infinite number of terms.
4Start with a square one unit by one unit
1
This is an example of an infinite series.
1
This series converges (approaches a limiting
value.)
Many series do not converge
5In an infinite series
a1, a2, are terms of the series.
an is the nth term.
Partial sums
nth partial sum
6Geometric Series
In a geometric series, each term is found by
multiplying the preceding term by the same
number, r.
7Geometric Series
Partial Sum of a Geometric Series Sn a
ar ar2 ar3 arn-1 -r Sn ar
ar2 ar3 arn Sn r Sn a arn
Sn (1 r) a (1 - rn)
8Sum of Converging Series
9 Power Series Using Calculator
10Example 1
11Example 2
12The partial sum of a geometric series is
The more terms we use, the better our
approximation (over the interval of convergence.)
13Example of a Power Series
14A power series is in this form
or
The coefficients c0, c1, c2 are constants.
The center a is also a constant.
(The first series would be centered at the origin
if you graphed it. The second series would be
shifted left or right. a is the new center.)
15Once we have a series that we know, we can find a
new series by doing the same thing to the left
and right hand sides of the equation.
This is a geometric series where r-x.
16Example 4
Given
find
We differentiated term by term.
17Example 5
Given
find
18Example 5
19This series would allow us to calculate a
transcendental function to as much accuracy as we
like using only pencil and paper!
p
20Convergent Series
Only two kinds of series converge 1)
Geometric whose r lt 1 2) Telescoping
series Example of a telescoping series the
middle terms cancel out
21Finding a series for tan-1 x
- 1. Find a power series that represents
on (-1,1) -
- Use integration to find a power series that
represents - tan-1 x.
- Graph the first four partial sums. Do the graphs
suggest convergence on the open interval (-1, 1)? -
- 4. Do you think that the series for tan-1 x
converges at x 1? -
22Guess the function