M38 Lec 010914
Transcript of M38 Lec 010914
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MATH 38Mathematical Analysis III
I. F. Evidente
IMSP (UPLB)
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Outline
1 Power Series
2 Power Series Representation of a Function
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Outline
1 Power Series
2 Power Series Representation of a Function
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DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
n=0
cn(xa)n = c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
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DefinitionLet {cn}n=0 be a sequence and a R. A power series in xa is a sum ofthe form
n=0
cn(xa)n = c0+ c1(xa)+ c2(xa)2+ ...+ cn(xa)n + ...
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Theorem
Letn=0
cn(xa)n be a power series. Exactly one of the following is true:
1 The power series converges only when x = a. (Case 1)2 The power series is absolutely convergent for all x R. (Case 2)3 There exists R > 0 such that the power series is absolutely convergent
for all x such that |xa| R. (Case 3)
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Theorem
Letn=0
cn(xa)n be a power series. Exactly one of the following is true:1 The power series converges only when x = a. (Case 1)2 The power series is absolutely convergent for all x R. (Case 2)3 There exists R > 0 such that the power series is absolutely convergent
for all x such that |xa| R. (Case 3)
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Remark1 This set of values is called the interval of convergence (IOC) of the
power series.
2 The radius of the interval of convergence is called the radius ofconvergence (ROC) of the power series.
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Remark1 This set of values is called the interval of convergence (IOC) of the
power series.2 The radius of the interval of convergence is called the radius of
convergence (ROC) of the power series.
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Outline
1 Power Series
2 Power Series Representation of a Function
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Consider the power series
n=0
xn
= 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:
We obtain convergent geometric series with r = x andn=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r =
x and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x
and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn =
1
1x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn =
1
1x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn = 1
1x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Consider the power series
n=0
xn = 1+x+x2+x3+ ...
Plug in x such that |x| < 1:We obtain convergent geometric series with r = x and
n=0
xn = 11x
If |x| < 1, the power seriesn=0
xn is just the same function as f (x)= 11x !
We sayn=0
xn is a power series representation of f (x)= 11x for |x| < 1.
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Definition
If a function f can be expressed as a power seriesn=0
cn(xa)n in someinterval I , then
we say that f is represented by the power series on I .That is,
f (x)=n=0
cn(xa)n for all x I
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Definition
If a function f can be expressed as a power seriesn=0
cn(xa)n in someinterval I , then we say that f is represented by the power series on I .
That is,
f (x)=n=0
cn(xa)n for all x I
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Definition
If a function f can be expressed as a power seriesn=0
cn(xa)n in someinterval I , then we say that f is represented by the power series on I .That is,
f (x)=n=0
cn(xa)n for all x I
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NoteThe following are synonymous:
1n=0
cn(xa)n is a power series representation (PSR) of f (x) on I .
2 The power series converges to f (x).3 The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
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NoteThe following are synonymous:
1n=0
cn(xa)n is a power series representation (PSR) of f (x) on I .2 The power series converges to f (x).
3 The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
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NoteThe following are synonymous:
1n=0
cn(xa)n is a power series representation (PSR) of f (x) on I .2 The power series converges to f (x).3 The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
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NoteThe following are synonymous:
1n=0
cn(xa)n is a power series representation (PSR) of f (x) on I .2 The power series converges to f (x).3 The sum of the power series is f (x).
RemarkNot all functions have a power series representation!
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Remember this!n=0
ar n = a1 r when |r | < 1
ExampleUsing the above, obtain a PSR for the following. Indicate for which valuesthe representation is true:
1 f (x)= 11+x
2 g (x)= 34x
3 h(x)= 2x3+2x2
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Remember this!n=0
ar n = a1 r when |r | < 1
ExampleUsing the above, obtain a PSR for the following. Indicate for which valuesthe representation is true:
1 f (x)= 11+x
2 g (x)= 34x
3 h(x)= 2x3+2x2
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Theorem (Differentiation of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then the function is differentiable for all |xa|
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Theorem (Differentiation of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then the function is differentiable for all |xa|
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Theorem (Differentiation of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then the function is differentiable for all |xa|
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Theorem (Differentiation of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then the function is differentiable for all |xa|
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Theorem (Differentiation of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then the function is differentiable for all |xa|
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Theorem (Differentiation of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then the function is differentiable for all |xa|
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Example
Given f (x)= 11x , find a PSR for f
(x) for |x| < 1.
ExampleFind a PSR for the following. Indicate the values of x for which the PSR istrue.
1 f (x)= 1(x+3)2
2 f (x)= x(2x)2
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Example
Given f (x)= 11x , find a PSR for f
(x) for |x| < 1.
ExampleFind a PSR for the following. Indicate the values of x for which the PSR istrue.
1 f (x)= 1(x+3)2
2 f (x)= x(2x)2
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Theorem (Integration of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then for all |xa|
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Theorem (Integration of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then for all |xa|
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Theorem (Integration of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then for all |xa|
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Theorem (Integration of a Power Series)Suppose that a function f is represented by a power series in xa that hasa nonzero radius of convergence R:
f (x)=n=0
cn(xa)n
Then for all |xa|
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Example
Use the fact that f (x)= 11x =
n=0
xn to find a PSR for ln(1x) for |x| < 1.
ExampleFind a PSR for the following. Indicate the values of x for which the PSR istrue.
1 f (x)= ln(x+1)2 f (x)= ln(23x)
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Example
Use the fact that f (x)= 11x =
n=0
xn to find a PSR for ln(1x) for |x| < 1.
ExampleFind a PSR for the following. Indicate the values of x for which the PSR istrue.
1 f (x)= ln(x+1)2 f (x)= ln(23x)
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Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx = ex +ex2
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Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx = ex +ex2
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Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx =
ex +ex2
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Example
To what function doesn=0
xn
n!converge to?
Remember this!
ex =n=0
xn
n!
Example
Find a PSR for coshx. Use the fact that coshx = ex +ex2
Power SeriesPower Series Representation of a Function