Ch 9.1 Power SeriesCalculus Graphical, Numerical, Algebraic byFinney, Demana, Waits, Kennedy

What 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.

For Example

• Power Series: an infinite sum of variables to a power.

• y = sin (x) can be represented as a power series:

Every time you add a term to the series it fits sin x even better.

Let’s check this out using the calculator and then the geometer sketchpad.

3 5 7 9x x x xsin x = x - + - + - ....

3! 5! 7! 9!

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-1 1 2 3 4 5 6 7

g x = x-x3

32

f x = sin x

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-1 1 2 3 4 5 6 7

g x = x-x3

32 +

x5

5432 -

x7

765432

f x = sin x

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-1 1 2 3 4 5 6 7

g x = x-x3

32 +

x5

5432 -

x7

765432 +

x9

98765432

f x = sin x

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-1 1 2 3 4 5 6 7

g x = x-x3

32 +

x5

5432 -

x7

765432 +

x9

98765432 -

x11

111098765432

f x = sin x

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-1 1 2 3 4 5 6 7

g x = x-x3

32 +

x5

5432 -

x7

765432 +

x9

98765432 -

x11

111098765432 +

x13

1312111098765432

f x = sin x

2.5

2

1.5

1

0.5

-0.5

-1

-1.5

-2

-1 1 2 3 4 5 6 7

g x = x-x3

32 +

x5

5432 -

x7

765432 +

x9

98765432 -

x11

111098765432 +

x13

1312111098765432 -

x15

15141312111098765432 +

x17

171615141312111098765432

f x = sin x

Power Series for cos x

If sin x can be represented by the power series:

3 5 7 9x x x xsin x = x - + - + - ....

3! 5! 7! 9!

Then cos x can be represented by the power series derived from taking the derivative of sin x:

2 4 6 8 2n

nx x x x xcos x = 1 - + - + - ....+(-1)

2! 4! 6! 8! 2n !

Let’s check it out on the calculator…

Power Series for cos x

Geometric Series

n

n

1 - rS = a

1 - r

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)

Sum of Converging Series

nn

nn

1 - rlim S = a , if r < 1, then r goes to

1 - r

zero and

a S =

No

1 - rif r > 1, the series diverges.

te: The interval of convergence is r < 1

Power Series Using Calculator

102

x = 1

2

To calculate a partial sum of a power series on the calculator,

x

you can find the expanded form by entering:

seq x , x, 1, 10

to get 1 4 9 16 2

2

5 36 49 64 81 100

and the sum by entering:

sum seq x , x, 1, 10

to get 385

Example of a Power Series

nn

n=0

2 3

1The function y = can be written as the power series:

1 - x

S = x which converges only for x < 1

this expands to 1 + x + x + x + ... this is an infinite series

that converges

1

n2

1

2

1to S = .

1 - x1

On your calculator, enter y = and 1 - x

y = sum seq x , n, 0, 20

Then test a value at home:

Enter y (.5)

Enter y (.5)

What do you notice?

Convergent Series

Only two kinds of series converge:

1) Geometric whose | r | < 1

2) Telescoping series

Example of a telescoping series: the middle terms cancel out

n 1 n = 1

n

1 1 1 = -

n n + 1 n n + 1

1 1 1 1 1 1 1 = 1 - + - + - + - +...

2 2 3 3 4 4 5

= sum of 1 - last term

= 1 + lim

1 -

n + 1

= 1 + 0 = 1

n 1

n 1

15

2

n 1

n 1

15

2

Graph both of these functions on your calculator!

Now graph both

y = ln (1+x)

And the power series below and check the fit!

Finding Power Series for Functions

Given that 1/(1-x) is represented by the power series:

1 + x + x2 + … + xn + …

On the interval (-1,1),

1. Find a power series that represent 1/(1+x) on (-1,1).

2. Find a power series that represents x/(1+x) on (-1,1).

3. Find a power series that represents 1(1-2x) on (- ½ , ½ )

4. Find a power series that represents

5. Find a power series that represents

and give its interval of convergence

1 1 = on (0,2)

x 1 + (x - 1)

1 1 1 =

3x 3 1 + (x - 1)

Finding Power Series for Functions

Given that 1/(1 - x) is represented by the power series:

1 + x + x2 + … + xn + … on the interval (-1,1),

1. Find a power series that represent 1/(1+x) on (-1,1)

1 - x + x2 – x3 … + (-x)n + …

2. Find a power series that represents x/(1+x) on (-1,1).

x – x2 + x3 – x4 + x5 – x6…. + (-1)n xn+1 + …

3. Find a power series that represents 1/(1 - 2x) on (- ½ , ½ )

1 + 2x + 4x2 + 8x3 + … + (2x)n + …

4. Find a power series that represents

1 – (x-1) + (x-1)2 – (x-1)3 + … + (-1)n (x-1)n + …

5. Find a power series that represents

1 1 = on (0,2)

x 1 + (x - 1)

1 1 1 =

3x 3 1 + (x - 1)

2 31 1 1 1 - (x-1) + (x-1) - (x-1) + ... on (0,2)3 3 3 3

Finding a series for tan-1 x

1. Find a power series that represents on (-1,1)

2. Use integration to find a power series that represents

tan-1 x.

3. 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?

21

(1 x )

Finding a series for tan-1 x

1. Find a power series that represents on (-1,1)

2. Use integration to find a power series that represents

tan-1 x.

3. Graph the first four partial sums. Do the graphs suggest convergence on the open interval (-1, 1)?

yes

4. Do you think that the series for tan-1 x converges at x = 1? Yes to

21

(1 x )2 4 6 n 2n1 - x + x - x + ...+ (-1) x + ...

3 5 7 2n 1n

2

1 x x x x dx = x - + - +...+ (-1) +...

1 x 3 5 7 2n 1

4

Guess the function

Define a function f by a power series as follows:

2 3 4 nx x x xf (x) = 1 + x + + + +...+ + ...

2! 3! 4! n!

Find f ‘(x).

What function is this?

Guess the function

Define a function f by a power series as follows:

2 3 4 nx x x xf (x) = 1 + x + + + +...+ + ...

2! 3! 4! n!

Find f ‘(x).

What function is this? ex

2 3 4 nx x x xf (x) = 1 + x + + + +...+ + ...

2! 3! 4! n!