Power 11.8a Series
Series of Powers of x [1] A power series convergent for 0 only [3] A power series convergent on the line Radius of Convergence [4] A power series convergent on finite interval
11.8a Power Series
1 1+ x
=
n =0
( -x)
xn n!
n
for - 1 < x < 1 for all real x
e =
x n =0
ln ( 1 + x ) = cos x =
( -1) n-1 x n
n =1 n
for - 1 < 1 x for all real x for all real x for all real x for all real x
(-1)
n =0
n x2 n (2 n )!
sin x =
n =0
(-1) n
x2 n (2 n )!
x 2 n+1 (2 n +1)!
cosh x =
n =0
sinh x =
n =0
x 2 n+1 (2 n +1)!
11.8a Power Series
c0 + c1 x + c2 x + c3 x + c4 x
2 3 4 x n =0
cn x
n
for - r < x < r
11.8a Questions About a Power Series
For what x does the P.S. converge? Does the P.S. converge absolutely or conditionally? Does the P.S. represent a familiar function?
11.8a Radius of Convrgnc; Ratio Test
Decide whether a n =
n =0 x
cn x n converges.
n =0
x an +1 cn +1 x n +1 cn +1 1 cn +1 L = lim = lim = x lim = , where = lim . n n * 8 n n an cn x cn R R n cn Then because x = L R , the RATIO TEST concludes absolute convergence for x < R, meaning - R < x < R , concludes and is divergence for x > R, meaning x < - R or x > R inconclusive for x = R, meaning x = R .
In the inconclusive case, the series may converge conditionally.
11.8a [1] Example of Ratio Test
Decide whether a
n =0 x n
=
c x
n=0 n
n
=
n !x
n =0
n
converges.
cn +1 1 (n + 1)! = lim = lim = lim (n + 1) =* n n * R n cn n! R = 0 Then because x = L R , the RATIO TEST concludes divergence for x > 0.
.
Clearly the series converges for x = 0 and no other x.
11.8a [3] Example of Ratio Test
Does a
n =0 x n
=
c x
n =0 n
n
=
( -1)
n=0
n
x2 n 22 n ( n!)2
converge ?
an +1 x 2 (n !) = 2 n+2 an 2 [(n + 1)!]2 x 2 n
2n+2
2n
2
x = 2 2
2
n! n + 1)! (
2
an +1 x2 1 L = lim = lim 2 = 0 . 2 n n an 2 ( n + 1) R = This series converges for all real x .
11.8a Radius of Convrgnc; Root Test
Decide whether a n =
n =0 x
cn x n converges.
n =0
L = lim
* n 8
n
an = lim cn x
n n
n
1 = x lim cn = , where = lim n cn . n R R n
n
x
Then because x = L R , the ROOT TEST concludes absolute convergence for x < R, meaning - R < x < R , concludes and is divergence for x > R, meaning x < - R or x > R inconclusive for x = R, meaning x = R .
In the inconclusive case, the series may converge conditionally.
11.8a [4] Power Series
( -3)n x n n =0 an+1 an n +1
=
3
n +1
x
n +1 n
n +1 n+2
3n x
=
3 x 1+ 1 n
2 1+ n
3x
The given series converges if 3 x < 1 OR x < 1 3 and diverges if 3 x > 1 OR x > 1 3 If x = 1 , the series becomes 3
11.8a [4] Power Series
If x =
1 3
, the series becomes =
n =0
x
( -3)n x n n +1
n =0
( -3) n (
1 )n 3
n +1
=
n =0
(m n 1) n +1
which fails to converge absolutely. If x = + 1 , the series alternates and 3 converges by the alternating series theorem. If x = - 1 , the series diverges. 3