2 - taylor series and convergence
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Lecture 10
-Taylor Series (Contd)
- Covergence of a series
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Newton Binomial
How to compute (1+x)^n ?
Newton Binomial
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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?
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Remainder term: important example
Try to do this!: find the n-order remainder ter of the following function!
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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!
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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
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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?
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Small oh
Strange rule:
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Illustration
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Confuse?
The following is wrong
But this one is correct!
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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,
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Taylor Formula for f
The Taylor Formula for f is
Example,
Find the Taylor Formula for
Find the Taylor Formula for arc tan x!
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Sequence and their limit
Consider the examples below
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Convergences of sequence
Example,
show that 1/n converge to 0!
Show that
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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.
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A little Exercise
Show that
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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!
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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
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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
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Geometric pictures
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Complex infinite series
Complex plane
Re
Im
L
Real infinite series
L L+L-
S0
S1S2
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Convergence of geometric series
If |r|
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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
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But
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is convergent
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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
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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
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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
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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
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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
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