2 - Taylor Series and Convergence

Lecture 10

-Taylor Series (Contd)

- Covergence of a series

Newton Binomial

How to compute (1+x)^n ?

Newton Binomial

The remainder term

Taylor polynomial will be most accurate when x is small

For this reason we define: error term or remainder term defined as

Rn f(x)= f(x)-Tn f(x)

Example,

What is the remainder term of any polynomial function?

Remainder term: important example

Try to do this!: find the n-order remainder ter of the following function!

Lagrange formula

Theorem: For some n+1 differentiable f on I, for any x in this interval, there is \zeta such that for 0 \leq \zeta \leq x or x \leq \zeta \leq 0, this expression is fulfilled

Without further explanation, I am sure youll be confused!

Estimate of remainder term

If we can find a constant M such that

Then the remainder term can be approximated as

Example, estimate e using n=8!

Estimate the error in estimating sin x = x

Limit when x-> 0

For any x near 0, according to Lagrange

Theorem: for any n+1 diff f, k= 0,1,2, .., n

Do u agree that the remainder is the smallest when x -> 0?

Small oh

Strange rule:

Illustration

Confuse?

The following is wrong

But this one is correct!

A bit strange but useful

Theorem, for any n+1 differentiable function f(x) and g(x),

Why?

Example

Compute T12 of f(x)=1/(1+x^2)!

Try to compute using g(t)=1/(1-t)

Once more,

Taylor Formula for f

The Taylor Formula for f is

Example,

Find the Taylor Formula for

Find the Taylor Formula for arc tan x!

Sequence and their limit

Consider the examples below

Convergences of sequence

Example,

show that 1/n converge to 0!

Show that

Limit of a function

Sandwich Theorem: of any a_n \leq b_n \leq c_n, if lim n \to \infty a_n and c_n = 0, so is for b_n

Theorem: using sandwich theorem, the following is true

Example, find lim t-> 0 cos (1/n) !

Show that converges to zero.

A little Exercise

Show that

Convergence of Taylor series

Taylor series is said to be convergent if

How to check?

Perform the above limit problem

Check if the remainder term is zero for n \to \infty

Example,

prove that the geometric series is convergent!

Prove that Tn(e^x) is convergent!

Infinite series Geometric series

If a = 1 and r= 1/2,

If a = 1 and r = 1

1+1+1+1+1+

If a = 1 and r = 1

1 1 + 1 1 + 1 1 +

If a = 1 and r = 2

1+2+4+8+16+

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= 1

diverges

diverges

diverges

Formal definition for convergence

Consider an infinite series The numbers ai may be real or complex.

Let Sn be the nth partial sum

The infinite series is said to be convergent if there is a number L such that, for every arbitrarily small > 0, there exists an integer N such that

The number L is called the limit of the infinite series.kshum 19

Geometric pictures

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Complex infinite series

Complex plane

Re

Im

L

Real infinite series

L L+L-

S0

S1S2

Convergence of geometric series

If |r|

Easy fact

If the magnitudes of the terms in an infinite series does not approach zero, then the infinite series diverges.

But the converse is not true.

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Harmonic series

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is divergent

But

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is convergent

Terminologies

An infinite series z1+z2+z3+ is called absolutely convergent if |z1|+|z2|+|z3|+ is convergent.

An infinite series z1+z2+z3+ is called conditionally convergent if z1+z2+z3+ is convergent, but |z1|+|z2|+|z3|+ is divergent.

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Examples

is conditionally convergent.

is absolutely convergent.

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Convergence tests

Some sufficient conditions for convergence.

Let z1 + z2 + z3 + z4 + be a given infinite series.

(z1, z2, z3, are real or complex numbers)

1. If it is absolutely convergent, then it converges.

2. (Comparison test) If we can find a convergent series b1 + b2 + b3 + with non-negative real terms such that

|zi| bi for all i,

then z1 + z2 + z3 + z4 + converges.

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http://en.wikipedia.org/wiki/Comparison_test

Convergence tests

3. (Ratio test) If there is a real number q < 1, such that

for all i > N (N is some integer),

then z1 + z2 + z3 + z4 + converges.

If for all i > N , , then it diverges

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http://en.wikipedia.org/wiki/Ratio_test

Convergence tests

4. (Root test) If there is a real number q < 1, such that

for all i > N (N is some integer),

then z1 + z2 + z3 + z4 + converges.

If for all i > N , , then it diverges.

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http://en.wikipedia.org/wiki/Root_test

Derivation of the root test from comparison test

Suppose that for all i N. Then

for all i N. But

is a convergent series (because q

Application

Given a complex number x, apply the ratio test to

The ratio of the (i+1)-st term and the i-th term is

Let q be a real number strictly less than 1, say q=0.99. Then,

Therefore exp(x) is convergent for all complex number x.

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Application

Given a complex number x, apply the root test to

The ratio of the (i+1)-st term and the i-th term is

Let q be a real number strictly less than 1, say q=0.99. Then,

Therefore exp(x) is convergent for all complex number x.

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Variations: The limit ratio test

If an infinite series z1 + z2 + z3 + , with all terms nonzero, is such that

Then

1.The series converges if < 1.

2.The series diverges if > 1.

3.No conclusion if = 1.

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Variations: The limit root test

If an infinite series z1 + z2 + z3 + , with all terms nonzero, is such that

Then

1.The series converges if < 1.

2.The series diverges if > 1.

3.No conclusion if = 1.

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Application

Let x be a given complex number. Apply the limit root test to

The nth term is

The nth root of the magnitude of the nth term is

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Useful facts

Stirling approximation: for all positive integer n, we have

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J0(x) converges for every x