Mathematical Methods for Physicists: A concise introduction - Site Map

INFINITE SERIES
(a) converges absolutely if R < 1;
(b) diverges if R > 1.
The test fails if R ˆ 1.
Gauss' test
If
u n‡1
u n
ˆ 1 G n ‡ c n
n 2 ;
where jc n j < P for all n > N, then the series P u n :
(a) converges (absolutely) if G > 1;
(b) diverges or converges conditionally if G 1.
Example A1.11
Consider the series 1 ‡ 2r ‡ r 2 ‡ 2r 3 ‡ r 4 ‡ 2r 5 ‡. The ratio test gives
u n‡1
ˆ 2jj;
r n odd
jj=2; r n even ;
u n
which indicates that the ratio test is not applicable. We now try the nth root test:
( p
p
n
2jr n j ˆ
np
2
n
jj; r n odd
ju n j ˆ p
n
jr n j ˆ jj; r n even
p
n
and so lim n!1 ju n j ˆ jj. r Thus if jrj < 1 the series converges, and if jrj > 1 the
series diverges.
Example A1.12
Consider the series
1 2
‡ 1 4 2
‡ 1 4 7 2
1 4 7 …3n 2†
‡‡ ‡:
3 3 6 3 6 9
3 6 9 …3n†
The ratio test is not applicable, since
u
lim
n‡1
n!1 ˆ lim
…3n ‡ 1†
2
n!1 …3n ‡ 3† ˆ 1:
u n
But Raabe's test gives
lim n 1 u
n‡1
n!1
u n
and so the series converges.
(
ˆ lim n 1 3n ‡ 1 ) 2
ˆ 4
n!1 3n ‡ 3 3 > 1;
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