The integral test revisited Statement of the test
The integral test revisited
Statement of the test
If f is a positive decreasing function then the infinite series and the integral
∞
X f ( k ) k =1 and
Z
∞ f ( x ) dx
1 either both converge or both diverge.
1.
Application of the integral test to the p -series:
I The series
∞
X k
− p k =1 converges for p > 1 and diverges for p ≤ 1.
I In particular, the harmonic series P
∞ k =1
1 k diverges.
Math 105 (Section 204)
2011W T2 1 / 6
Exercise
For which values of p do the following series converge?
∞
X k =10
1 k ln k [ln(ln k )] p
A.
p > 1
B.
p > 0
C.
p ≤ 0
D.
p < − 1
Math 105 (Section 204)
2011W T2 2 / 6
The ratio test
Statement
Let P k a k be an infinite series with positive terms. Let r = lim k →∞ a k +1
.
a k
1.
If 0 ≤ r < 1, the series converges.
2.
If r > 1 (including r = ∞ ), the series diverges.
3.
If r = 1, the test is inconclusive.
Math 105 (Section 204)
2011W T2 3 / 6
The ratio test vs the integral test
The ratio test is useful for series the form a k or k !, e.g.,
∞
X
P
( k k a
!) k
3 where a k involves terms of k =1
(3 k )!
for which the integral test does not provide a solution (why?). It is however less useful if a k involves polynomials in k , such as
∞
X
1 k p
.
k =1
Here the ratio test is inconclusive, but the integral test works!
Math 105 (Section 204)
2011W T2 4 / 6
Comparison tests
Basic comparison test
Let
1.
P k
If 0 a
< k a k and P k b k be series with positive terms.
≤ b k and P b k converges, then P a k converges.
2.
If 0 < b k
≤ a k and P b k diverges, then P a k diverges.
Limit Comparison Test
Let P k a k and P k b k be series with positive terms, and lim k →∞ a k b k
= L .
1.
If 0 < L < ∞ , then P a k and P b k both converge or both diverge.
2.
If L = 0 and P b k converges, then P a k converges.
3.
If L = ∞ and P b k diverges, then P a k diverges.
Math 105 (Section 204)
2011W T2 5 / 6
Test your testing skills
Which of the following series converge?
∞
X k =1 k
−
1 k
,
∞
X k =1
2 k k !
, k k
∞
X tan k =1
1 k
1.
first
2.
second
3.
third
4.
first and second
5.
first and third
Math 105 (Section 204)
2011W T2 6 / 6
