seriesconv.htm Power Series Convergence
Math 181 Calculus II Power Series Convergence JL


Convergence and Absolute Convergence

·
å*¥CN CONVerges iff



lim
N ® ¥ 
 N
å
 * 
 Cn
exists (finite).

Compare to

·
ò*¥f(x)  dx CONVerges iff



lim
X ® ¥ 
 ó
õ 		X

* 
 f(x)  dx
exists (finite).

·
å*¥Cn CONVerges ABSsolutely iff å*¥|Cn| converges or



lim
N ® ¥ 
 N
å
 * 
 |Cn|
exists (finite).

Compare to
·
ò*¥f(x)  dx CONVerges ABSsolutely iff ò*¥|f(x)|  dx converges or



lim
X ® ¥ 
 ó
õ 		X

* 
 |f(x)|  dx
exists (finite).

·
å*¥|Cn| converges iff the sequence å*N |Cn| is bounded.
·
ò*¥|f(x)| converges iff the function ò*X |f(x)|  dx is bounded (as X ® ¥).


Comparison Test for Absolute Convergence

·
If


0 £ An £ Bn,
then


0 £ ¥
å
 * 
 An £ ¥
å
 * 
 Bn.
So that if the bigger series å*¥Bn CONVerges, the smaller series å*¥An CONVerges also.

If the smaller series å*¥An DIVerges, the bigger series å*¥An DIVerges also.

·
If


0 £ f(x) £ g(x),
then


0 £ ó
õ 		¥

* 
 f(x)  dx £ ó
õ 		¥

* 
 g(x)  dx.
So that if the bigger integral ò*¥g(x)  dx CONVerges, the smaller integral ò*¥f(x)  dx CONVerges also.

If the smaller integral ò*¥f(x)  dx DIVerges, the bigger integral ò*¥g(x)  dx DIVerges also.


Ratio Test for ABSolute CONVergence

For the series å*¥CN, suppose that



lim
n ® ¥ 
 ê
ê
ê 		Cn+1
Cn
 ê
ê
ê 		= L.

·
If 0 £ L < 1, the series å*¥CN CONVerges ABSolutely.
·
If 1 < L £ ¥, the series å*¥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 ån=0¥ an xn, there is a number R, 0 £ R £ ¥ for which


¥
å
 n=0 
 an xn ì
í
î
CONVerges ABSolutely for |x| < R,
DIVerges for |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 ån=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 ån=1¥ n an xn-1:
If


f(x) = ¥
å
 n=0 
 an xn, |x| < R,
then


f¢(x) = ¥
å
 n=1 
 n an xn-1 = ¥
å
 n=0 
 (n+1) an+1 xn, |x| < R,
and


ó
õ 		x

0 
 f(t)  dt = ¥
å
 n=0 
 an
n+1
 xn+1 = ¥
å
 n=1 
 an-1
n
 xn, |x| < R



File translated from TEX by TTH, version 2.86.
On 18 Jan 2001, 09:23.