Convergence and Divergence
Summary of Convergence and Divergence Tests for Series
TEST nth-term Geometric series p-series Integral SERIES ∑ an CONVERGENCE OR DIVERGENCE Diverges if lim n→∞ an ≠ 0 (i) Converges with sum S = (ii) Diverges if r ≥ 1
1
p
COMMENTS Inconclusive if lim n →∞ an = 0 Useful for the comparison tests if the nth term an of a series is similar to arn-1 Useful for the comparison tests if the nth term an of a series is similar to 1/np The function f obtained from an = f ( n ) must be continuous,
∑ ar n =1
∞
n −1
a if r < 1 1− r
∑n n =1
∞
(i) Converges if p > 1 (ii) Diverges if p ≤ 1 (i) Converges if (ii) Diverges if n ∑a n =1
∞
n
∫ f ( x ) dx converges
1 ∞ 1
∞
an = f ( n ) Comparison an > 0, bn > 0
∫ f ( x ) dx diverges
∑ a ,∑b n (i) If
∑b
n
converges and an ≤ bn for every n, then
∑a n n
positive, decreasing, and readily integrable. The comparison series ∑ bn is often a geometric series of a pseries. To find bn in (iii), consider only the terms of an that have the greatest effect on the magnitude. Inconclusive if L=1 Useful if an involves factorials or nth powers If an>0 for every n, the absolute value sign may be disregarded. Inconclusive if L=1 Useful if an involves nth powers If an>0 for every n, the absolute value sign may be disregarded. Applicable only to an alternating series Useful for series that contain both positive and negative terms
converges. (ii) If ∑ bn diverges and an ≥ bn for every n, then diverges.
(iii) If lim n→∞ ( an / bn ) = c > 0 , them both series converge or both Ratio
∑a
diverges.
∑a
n
an +1 = L (or ∞), the series n →∞ a n (i) converges (absolutely) if L1 (or ∞)
If lim If lim n→∞ n Root
∑a
n
an = L (or ∞), the series
(i) converges (absolutely) if L1 (or ∞) Alternating series
∑ ( −1)
n
an
Converges if ak ≥ ak +1 for every k and lim n →∞ an = 0 If
∑a
n
∑a
an > 0 n ∑a
n
converges, then
∑a
n
TEST nth-term Geometric series p-series Integral SERIES ∑ an CONVERGENCE OR DIVERGENCE Diverges if lim n→∞ an ≠ 0 (i) Converges with sum S = (ii) Diverges if r ≥ 1
1
p
COMMENTS Inconclusive if lim n →∞ an = 0 Useful for the comparison tests if the nth term an of a series is similar to arn-1 Useful for the comparison tests if the nth term an of a series is similar to 1/np The function f obtained from an = f ( n ) must be continuous,
∑ ar n =1
∞
n −1
a if r < 1 1− r
∑n n =1
∞
(i) Converges if p > 1 (ii) Diverges if p ≤ 1 (i) Converges if (ii) Diverges if n ∑a n =1
∞
n
∫ f ( x ) dx converges
1 ∞ 1
∞
an = f ( n ) Comparison an > 0, bn > 0
∫ f ( x ) dx diverges
∑ a ,∑b n (i) If
∑b
n
converges and an ≤ bn for every n, then
∑a n n
positive, decreasing, and readily integrable. The comparison series ∑ bn is often a geometric series of a pseries. To find bn in (iii), consider only the terms of an that have the greatest effect on the magnitude. Inconclusive if L=1 Useful if an involves factorials or nth powers If an>0 for every n, the absolute value sign may be disregarded. Inconclusive if L=1 Useful if an involves nth powers If an>0 for every n, the absolute value sign may be disregarded. Applicable only to an alternating series Useful for series that contain both positive and negative terms
converges. (ii) If ∑ bn diverges and an ≥ bn for every n, then diverges.
(iii) If lim n→∞ ( an / bn ) = c > 0 , them both series converge or both Ratio
∑a
diverges.
∑a
n
an +1 = L (or ∞), the series n →∞ a n (i) converges (absolutely) if L1 (or ∞)
If lim If lim n→∞ n Root
∑a
n
an = L (or ∞), the series
(i) converges (absolutely) if L1 (or ∞) Alternating series
∑ ( −1)
n
an
Converges if ak ≥ ak +1 for every k and lim n →∞ an = 0 If
∑a
n
∑a
an > 0 n ∑a
n
converges, then
∑a
n