Sequences and Series& Taylor series

Patrick DoyleElwin Martin

Charles Ye

Series and Sequences

• Sequence: function whose domain is a set of all positive integers (n=1,2,3…)

Limit of a Series

lim nna L

example

lim n

ne

lim 0n

ne

nth term test

• If o or if the limit does not exist, then the series diverges.

• Apples to any series • Cannot be used to prove convergence

example

• Not enough information

» Series diverges

1

1

n n

2

1

21

n n

Geometric series

• Converges if the absolute value of r is less than one

• Diverges if the absolute value of r is greater than or equal to one

Where a is the first time

1

n

n

ar

1

aS

r

example2

1

2

2

3

2 1 2

2 3 4 3

2 12 1

1 2 3 2

n

nn

nn

n

Converges

S

Alternating harmonic series

Harmonic series

Telescoping series

• Series where the later terms start to cancel out

• Converges to the numbers that do not cancel out

example

Converges to 1

Integral Test

example

2 21 1

2

2

2

lim1 1

1

2

/ 2

1 1 1lim lim ln( 1)

2 21

lim [ln( 1) ln(2)]2

N

Nn

n n

n

n ndn

n n

u n

du ndn

du n dn

dn nn

n

diverges

P-series test

• If p is greater than one, the series converges• If p is equal or less than one, the series

diverges

Direct Comparison Test

Converges if and converges

Diverges if and diverges

example

21

2 2

cos

cos 1n

n

n

n

n nconverges

Limit Comparison Test

If is a number greater than zero and converges, then

converges

If is a number greater than zero and diverges, then

diverges

-Both series a and b must be greater than zero- The limit must not be infinity, zero, or negative

Root Test

There exists r such that if r < 1, then the series converges. If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

lim nn

na r

example

1

5

3

5lim 5

3

n

n

n

n

n

n

ndiverge

Ratio Test

•There exists a number r such that: if r < 1, then the series converges; If r > 1, then the series diverges. If r = 1, the ratio test is inconclusive, and the series may converge or diverge.

Error in Alternating series

1

Or in simple terms, the error of a sum is less than or equal

to the first neglected term.

This is useful for two things: finding an approximate erro

n nError of S a

r

for a given point and for finding the term at which the error

is within some bound.

Radius and Interval of Convergence

• In some instances there are series written in terms of x such as and in such cases there are a range of x values for which the series converges.

• The radius of curvature and interval of convergence are found by applying the ratio test and setting the ratio to less than one and solving for x.

• Once the interval has been found, check the end points.

1 !n

n

n

x

example

11 of interval and 1 of Radius

.n as convergeneither

since divergeboth 1&1

1

lim

00

1

0

x

x

x

x

x

n

n

n

n

n

n

n

n

n

Power Series

• Representation of a function as a summation of infinitely many polynomials.

C= x value at which the series is centered

Finding the Coefficient

I FOUND IT!!!

• Using this, you can plug it in for several terms then find a the equation for the general term!

• Now you have a Taylor Series!

(Actually finding the coefficient)2 3

0 1 2 30

0

21 2 3

1

22 3 4

2

( ) ...

(0) ,

'( ) 2 3 ...

'(0)

''( ) 2 6 12 ...

''(0)

2

(0)

!

nn

n

n

n

f x a x a a x a x a x

f a

f x a a x a x

f a

f x a a x a x

fa

fa

n

Example

Find the first 3 terms and the general term of the Taylor series for sin(x) centered at 0.

f (x) = sin(x) f (0) = 0

f '(x) = cos(x) f '(0) = 1

f '' = -sin(x) f ''(0) = 0

f ''' = -cos (x) f '''(0) = -1

f '''' = sin(x) f ''''(0) = 0

f''''‘=cos(x) f''''‘(0)=1

Manipulating the Series

• You can tweak the formulas you memorized to make finding the Maclaurin Series for many more equations much faster and EASIER. So do that.

• You can multiply the whole series by a number, substitute for x, or take the integral or derivative.

Maclaurin Series

• A Maclaurin series is a Taylor Series that centers at 0 (i.e. c=0).

• Here are some to memorize:

2 3

0

11 ... ...for 1

1n n

n

x x x x x xx

Example

Find the first four terms of the Maclaurin series for 5e2x.

Lagrange Error Bound1 1( )( )

( 1)!

n nn

f aR x x cn

Where f(x) is centered at c and a is some number between x and c.