Mathematical Methods for Physicists: A concise introduction - Site Map

INFINITE SERIES
Using the ratio test, we have
u n‡1
u n
so the series converges.
ˆ
1
…n ‡ 1†! 1 n! ˆ n!
…n ‡ 1†! ˆ 1
n ‡ 1 < 1;
Integral test.
If f …x† is positive, continuous and monotonic decreasing and is such that
f …n† ˆu n for n > N, then P u n converges or diverges according as
Z 1
Z M
f …x†dx ˆ lim f …x†dx
N
M!1 N
converges or diverges. We often have N ˆ 1 in practice.
To prove this test, we will use the following property of de®nite integrals:
If in a x b; f …x† g…x†, then
Z b
Now from the monotonicity of f …x†, we have
a
f …x†dx
Z b
a
g…x†dx.
u n‡1 ˆ f …n ‡ 1† f …x† f …n† ˆu n ; n ˆ 1; 2; 3; ...:
Integrating from x ˆ n to x ˆ n ‡ 1 and using the above quoted property of
de®nite integrals we obtain
u n‡1
Z n‡1
Summing from n ˆ 1toM 1,
n
f …x†dx u n ; n ˆ 1; 2; 3; ...:
u 1 ‡ u 2 ‡‡u M
Z M
1
f …x†dx u 1 ‡ u 2 ‡‡u M1 :
…A1:3†
If f …x† is strictly
R
decreasing, the equality sign in (A1.3) can be omitted.
M
If lim M!1 1
f …x†dx exists and is equal to s, we see from the left hand inequality
in (A1.3) that u 1 ‡ u 2 ‡‡u M is monotonically increasing and bounded
above by s, so that P R M
u n converges. If lim M!1 1
f …x†dx is unbounded, we see
from the right hand inequality in (A1.3) that P u n diverges.
Geometrically, u 1 ‡ u 2 ‡‡u M is the total area of the rectangles shown
shaded in Fig. A1.3, while u 1 ‡ u 2 ‡‡u M1 is the total area of the rectangles
which are shaded and non-shaded. The area under the curve y ˆ f …x† from x ˆ 1
to x ˆ M is intermediate in value between the two areas given above, thus illustrating
the result (A1.3).
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