Mathematical Methods for Physicists: A concise introduction - Site Map

APPENDIX 1 PRELIMINARIES
Example A1.1.
Show that the series
X 1
nˆ1
is convergent and has sum s ˆ 1.
1
2 n ˆ 1
2 ‡ 1 2 2 ‡ 1 2 3 ‡
Solution:
then
Let
Subtraction gives
1 1
2
s n ˆ 1
2 ‡ 1 2 2 ‡ 1 2 3 ‡‡ 1 2 n ;
1
2 s n ˆ 1
2 2 ‡ 1 2 3 ‡‡ 1
2 n‡1 :
s n ˆ 1
2 1
2 n‡1 ˆ 1
2 1 1 2 n
; or s n ˆ 1 1 2 n :
Then since lim n!1 s n ˆ lim n!1 …1 1=2 n †ˆ1, the series is convergent and has
the sum s ˆ 1.
Example A1.2.
Show that the series P 1
nˆ1 …1†n1 ˆ 1 1 ‡ 1 1 ‡ is divergent.
Solution: Here s n ˆ 0 or 1 according as n is even or odd. Hence lim n!1 s n does
not exist and so the series is divergent.
Example A1.3.
Show that the geometric series P 1
nˆ1 arn1 ˆ a ‡ ar ‡ ar 2 ‡; where a and r are
constants, (a) converges to s ˆ a=…1 r† if jrj < 1; and (b) diverges if jrj > 1.
Solution:
Then
Let
Subtraction gives
s n ˆ a ‡ ar ‡ ar 2 ‡‡ar n1 :
rs n ˆ ar ‡ ar 2 ‡‡ar n1 ‡ ar n :
…1 r†s n ˆ a ar n or s n ˆ a…1 rn †
:
1 r
512