∑*∞Cn CONVerges ABSsolutely iff
∑*∞|Cn| converges or
lim
N → ∞
N ∑ *
|Cn|
exists (finite).
Compare to
•
∫*∞f(x) dx CONVerges ABSsolutely iff
∫*∞|f(x)| dx converges or
lim
X → ∞
⌠ ⌡
X
*
|f(x)| dx
exists (finite).
•
∑*∞|Cn| converges iff the sequence ∑*N |Cn| is bounded.
•
∫*∞|f(x)| converges iff the function ∫*X |f(x)| dx is bounded (as X → ∞).
Comparison Test for Absolute Convergence
•
If
0 ≤ An ≤ Bn,
then
0 ≤
∞ ∑ *
An ≤
∞ ∑ *
Bn.
So that if the bigger series ∑*∞Bn CONVerges, the
smaller series ∑*∞An CONVerges also.
If the smaller series ∑*∞An DIVerges, the
bigger series ∑*∞An DIVerges also.
•
If
0 ≤ f(x) ≤ g(x),
then
0 ≤
⌠ ⌡
∞
*
f(x) dx ≤
⌠ ⌡
∞
*
g(x) dx.
So that if the bigger integral ∫*∞g(x) dx CONVerges, the
smaller integral ∫*∞f(x) dx CONVerges also.
If the smaller integral ∫*∞f(x) dx DIVerges, the
bigger integral ∫*∞g(x) dx DIVerges also.
Ratio Test for ABSolute CONVergence
For the series ∑*∞CN, suppose that
lim
n → ∞
⎢ ⎢
Cn+1
Cn
⎢ ⎢
= L.
•
If 0 ≤ L < 1, the series ∑*∞CN
CONVerges ABSolutely.
•
If 1 < L ≤ ∞, the series ∑*∞CN
DIVerges.
•
If L = 1, we are not sure - additional
information is needed about DIVergence or CONVergence and/or ABS0lute
CONVergence.
Power Series, Radius of Convergence, and Interval of Convergence
•
For a power series ∑n=0∞ an xn, there is a number
R, 0 ≤ R ≤ ∞ for which
∞ ∑ n=0
an xn
⎧ ⎪ ⎨
⎪ ⎩
CONVergesABSolutelyfor |x| < R,
DIVergesfor |x| > R.
The number R is called the radius of convergence of
the power series. R can often be determined by the Ratio Test.
•
If the power series ∑n=0∞ an xn, converges for x = x0, then for all x, |x| < |x0| the power series
CONVerges ABSolutely. Thus the radius of convergence ,
R,is greater than or equal |x0|.
•
If f(x) is represented by a convergent power series for
|x| < R, then for |x| < R, its derivative is represented by the
convergent series ∑n=1∞ n an xn−1:
If
f(x) =
∞ ∑ n=0
an xn, |x| < R,
then
f′(x) =
∞ ∑ n=1
n an xn−1 =
∞ ∑ n=0
(n+1) an+1 xn, |x| < R,
and
⌠ ⌡
x
0
f(t) dt =
∞ ∑ n=0
an
n+1
xn+1 =
∞ ∑ n=1
an−1
n
xn, |x| < R
File translated from
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TTH,
version 4.03. On 22 Feb 2012, 13:24.