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Power
series
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Power
series or polynomial with infinitely many terms |
The
sum of a power series is a function |
Maclaurin and Taylor series |
The radius of convergence or the interval of convergence |
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Power
series or polynomial with infinitely many terms |
A
real power series in x
around the origin (or centered at the origin) is a series of functions
of the form |
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and
the power series around a given point x
= x0 (or centered
at x0)
is a series of the form |
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where
the coefficients an
are fixed real numbers and x
is a real variable. |
A
power series with real coefficients is said to be real or complex
according as both x
and x0
are real or complex
numbers. |
Therefore,
the nth
partial sum of a power series is a polynomial of degree n, |
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The
sum of a power series is a function |
The
sum of a power series is a function |
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the domain of
which
is the set of those values of x
for which the series converges to the value of the function. |
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Maclaurin
and Taylor series |
Consider
the polynomial function |
f(x) = anxn
+ an -
1xn -
-
1 + · · · + a3x3
+ a2x2 + a1x
+ a0.
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If
we write the value of the function and the values of its successive
derivatives, at the origin, then |
f(0)
=
a0, f '(0) = 1· a1,
f ''(0) = 1· 2a2,
f '''(0) = 1· 2· 3a3, . . .
, f (n)(0) =
n!an |
so we
get the coefficients; |
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Then,
the polynomial
f(x)
with infinitely many terms, written as the power series |
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and |
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where
0! = 1,
f (0)(x0)
= f(x0)
and
f (n)(x0)
is the nth
derivative of f at
x0, |
represents an infinitely differentiable function and is called Maclaurin
series and Taylor series respectively. |
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The radius of convergence or the interval of
convergence |
- If the power series anxn
converges when x = x1,
then it converges for every x
that is closer to the |
origin than x1,
that is, whenever | x
| < | x1 |.
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- If the power series anxn
diverges when x = x1,
then it diverges for every x
that is further from the origin |
than x1,
that is, whenever | x
| > | x1 |. |
The
real power series converges for all absolute value of x
that are less than a number r,
called the radius of convergence or the
interval of convergence,
written | x |
< r that is, the open interval
-
r < x < r. |
We
apply the root test or the ratio test to find the interval of
convergence. Thus, |
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the power series |
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converges, |
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the power series diverges. |
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Therefore
the inequality, |
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defines the interval
in which the power series is absolutely convergent. |
Thus,
denoting the right side of the above inequality by r,
we get
the interval of convergence
| x | < r
saying, for
every x
between -
r
and r
the series
converges absolutely while, for
every x
outside that interval the series
diverges.
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Therefore,
the radius of convergence of the power series, |
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The power series
anxn
converges absolutely at every point x
from the open interval -
r < x < r and
diverges for all
x outside this interval. At
the endpoints -
r
and r,
the series may converge or diverge so these
points must be checked for convergence, individually. |
The power series
an(x
-
x0)n
converges absolutely at every point x
from the open interval |x
-
x0|
< r or |
x0
-
r < x < x0
+ r and
diverges for all
x outside this interval.
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Note
that the series may or may not converge when | x
-
x0 | = r
that is, when x
=
x0
-
r
or x
=
x0
+
r.
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To
determine whether the power series converge or diverge at endpoints we
should plug each endpoint into the
given series and apply appropriate test for convergence. |
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Functions
contents E |
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