Convergence Tests for Series

Background. We say that A converges absolutely if ∑_{k=k_{0}}^{∞} |a_{k}| converges. If A converges absolutely, then A converges. If A converges but does not converge absolutely, then we say that A converges conditionally. If A does not converge (to a finite number), then we say that A diverges.

Context. Let A = ∑ a_{n} := ∑_{n=n_{0}}^{∞} a_{n}. Let B = ∑_{n=n_{0}}^{∞} b_{n}. Let f(n) = a_{n}. (f is a function which agrees with the sequence (a_{n})_{n=1}^{∞}.) Let I = ∫_{x_{0}}^{∞} f(x) dx.

Convergence Test  Assumptions  Conclusion (Does A converge?)
Book ref.  Test name  abs. other conditions  converges diverges inconclusive
§10.2 p437 limit of terms  no (a_{n} ↛ 0)  N/A  if a_{n} ↛ 0 if a_{n} → 0
§10.3 p444 integral comp.  yes f cont., +, ↘  A iff I  (Ā iff Ī) never
§10.4 p448 ordinary comp. yes 0 ≤ a_{k} ≤ b_{k}  A if B  (B̄ if Ā) otherwise
§10.4 p449 limit comp.  yes \frac{a_{n}}{b_{n}} → L ≠ 0:  A iff B  (Ā iff B̄) −
\frac{a_{n}}{b_{n}} → L = 0:  A if B  (B̄ if Ā) otherwise
\frac{a_{n}}{b_{n}} → ∞:  (B if A) Ā if B̄  otherwise
§10.5 p455 ratio  yes |\frac{a_{n+1}}{a_{n}} | → ρ  if ρ < 1 if ρ > 1 if ρ = 1
(not in book) n-th root test  yes ^{n} \sqrt{a_{n}} → ρ  if ρ < 1 if ρ > 1 if ρ = 1
§10.5 p454 alternating series NO alternating, |a_{n}| ↘ if a_{n} → 0 if a_{n} ↛ 0 −

f ↘ means that f is a nonincreasing function ( f(x) ≥ f(y) if x < y). (Parenthesized statements can be inferred from the other information in the table.) The column labeled abs. indicates whether the test assumes that the terms are positive (in which case the test can only be used to show absolute convergence).

Test name  When used

limit of terms Used to show divergence. Can’t use to show convergence. integral comparison test Usually used to show absolute convergence when the formula for the terms of the

series is a function you can integrate.

ordinary comparison Used to show absolute convergence (respectively divergence) by comparing with a

simpler series whose terms are larger (respectively smaller).

limit comparison Used to show absolute convergence or diverence by comparing with a simpler series

(e.g. a p-series).

ratio and n-th root Used to show absolute convergence. Used most often for series with powers and factorials (e.g. to find the radius of convergence of a power series). Can’t use for a function that involves only addition, subtraction, multiplication, division, roots, or (constant) powers, because it will always be inconclusive.

alternating series Used most often to show convergence at an endpoint of the interval of convergence of a power series.

Example of Series  When does it converge?
Name  Series  converges diverges inconclusive
p-series  ∑_{n=0}^{∞} \frac{1}{n^{p}}  if p > 1  if p ≤ 1
geometric series ∑_{n=0}^{∞} x^{n} = \frac{1}{1−x}  if |x| < 1 if |x| ≥ 1
power series ∑_{n=0}^{∞} c_{n}x^{n} if |x| < ρ if |x| > ρ if |x| = ρ
collapsing sum ∑_{n=0}^{∞}(a_{n+1} − a_{n}) = −a_{0} if a_{n} → 0 if a_{n} ↛ 0