September 6, 2011 15:41
c09
Sheet number 50 Page number 645
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Summary of Convergence Tests
name
Divergence Test
(9.4.1)
Integral Test
(9.4.4)
statement
If lim uk ≠ 0, then
comments
If lim uk = 0, then
k→ + ∞
may not converge.
uk diverges.
k→+ ∞
Let
uk be a series with positive terms. If f is a function
that is decreasing and continuous on an interval [a, + ∞)
and such that uk = f (k) for all k ≥ a, then
∞
k =1
uk
+∞
and
uk may or
This test only applies to series that
have positive terms.
Try this test when f (x) is easy to
integrate.
f (x) dx
a
both converge or both diverge.
∞
Let
a and
k=1 k
terms such that
Comparison Test
(9.5.1)
∞
b
k=1 k
be series with nonnegative
a1 ≤ b1, a2 ≤ b2, . . . , ak ≤ bk , . . .
If
bk converges, then
ak converges, and if
diverges, then
bk diverges.
Let
ak and
ak
bk be series with positive terms and let
a
r = lim k
k→ + ∞ bk
Limit Comparison Test
(9.5.4)
If 0 < r < + ∞, then both series converge or both diverge.
Let
Ratio Test
(9.5.5)
Root Test
(9.5.6)
Try this test as a last resort; other
tests are often easier to apply.
This is easier to apply than the
comparison test, but still requires
some skill in choosing the series
bk for comparison.
uk be a series with positive terms and suppose that
r=
uk+1
k→+ ∞ uk
lim
(a) Series converges if r < 1.
(b) Series diverges if r > 1 or r = + ∞.
(c) The test is inconclusive if r = 1.
Let
This test only applies to series with
nonnegative terms.
Try this test when uk involves
factorials or kth powers.
uk be a series with positive terms and suppose that
r=
lim
k→+ ∞
k
uk
(a) The series converges if r < 1.
(b) The series diverges if r > 1 or r = + ∞.
(c) The test is inconclusive if r = 1.
Try this test when uk involves kth
powers.
If ak > 0 for k = 1, 2, 3, . . . , then the series
Alternating Series Test
(9.6.1)
a1 − a 2 + a 3 − a 4 + . . .
−a1 + a2 – a3 + a4 − . . .
converge if the following conditions hold:
(a) a1 ≥ a2 ≥ a3 ≥ . . .
(b) lim ak = 0
This test applies only to alternating
series.
k→+ ∞
Let
Ratio Test for
Absolute Convergence
(9.6.5)
uk be a series with nonzero terms and suppose that
| uk +1 |
k→+ ∞ | uk |
r = lim
(a) The series converges absolutely if r < 1.
(b) The series diverges if r > 1 or r = + ∞.
(c) The test is inconclusive if r = 1.
The series need not have positive
terms and need not be alternating
to use this test.