nth-Term
? ? Ann=1Diverges: lim An ? 0 n??Cannot be used to show convergence.
Geometric Series
? ? Ar?n=0Diverges: |r| ? 1Converges: |r| < 1Sum: S= a/(1-r)
p-Series
? ? 1/(n^p)n=1Diverges: p ? 1Converges: p > 1
Integral
? ? Ann=1An = ƒ(n) ? 0Diverges: ? ƒ(x) dx divegresConverges: ? ƒ(x) dx converges
Root
? ? Ann=1Diverges: lim |(An+1)/(An)| > 1 n??Converges: lim |(An+1)/(An)| < 1 n??If = 1, test is inconclusive.
Ratio
? ? Ann=1Diverges: lim ??(|An|) > 1 n??Converges: lim ??(|An|) < 1 n??If = 1, test is inconclusive.
Direct Comparison
? ? Ann=1Diverges: 0 < Bn ? An and ? Bn divergesConverges: 0 < An ? Bn and ? Bn converges
Limit Comparison
? ? Ann=1Diverges: lim (An)/(Bn) = L > 0 n?? and ? Bn divergesConverges: lim (An)/(Bn) = L > 0 n?? and ? Bn converges
Nth Term Test
if the limit of An as n approaches infinity does not equal zero, diverges
Nth Term Diverges
limit does not equal zero
Nth Term Converges
Nth term test cannot be used to show convergence
Geometric Series
Form ar^(n)
Geometric Series converges
absolute value of r is less than 1
Geometric Series diverges
absolute value of r is greater than or equal to 1
Sum of a geometric series
a/(1-r) where a is the first term
Telescoping Series
lim b sub n = L (exists)
Sum of telescoping series
b sub 1 - L = sum
P Series
1/n^p
P series converges
p is greater than 1
P series diverges
p is less than or equal to 1
Alternating series
(-1)^n(a sub n)
Alternating series converges
0 is less than a sub n +1 which is less than a sub n
Integral test
converges if the integral converges, diverges if the integral diverges
Root test
take the nth root of the absolute value of A sub n
Root test converges
if the result is less than 1
Root test diverges
if the result is greater than 1
Ratio test converges
if the result is less than 1
Ratio test diverges
if the result is greater than 1
Direct comparison converges
if the original is between the new and zero, adn the new one converges
Direct comparison diverges
if the original is greater than the new, which diverges
Limit comparison converges
if the limit is greater than 0 (exists) and the b sub n converges
Limit comparison diverges
if the limit is greater than 0 (exists) and the b sub n diverges
Nth term test (Divergent)
Series: ? a(n)*Condition*:lim (a) ? 0
Geometric Series (Convergent)
Series: ? a[r^n] *<-- n = 0**Condition*:0 < |r| < 1
Geometric Series (Divergent)
Series: ? a[r^n] *<-- n = 0**Condition*:|r| ? 1
Geometric Series (Sum)
Series: ? a[r^n] *<-- n = 0*S = __a__.
.....
.(1-r)
Telescoping Series (Convergent)
Series: ? [b(n) - b(n+1)]*Condition*:lim b(n) = Lb(1) - L
P-Series (Convergent)
Series: ? [1/(n^p)]*Condition*:P >1
P-Series (Divergent)
Series: ? [1/(n^p)]*Condition*:P ? 1
Alternating Series (Convergent)
Series: ? [(-1)^(n-1)] a(n)*Conditions*:0 < a(n+1) ? a(n) ...AND: lim a(n) = 0
Alternating Series (Remainder)
*Remainder*|S-S(N)| = | R(N) | ? a (N+1)
Integral test (Convergent/Divergent)
Series: ? a(n)*Conditions*:positive: a(n) = f(n) > 0..
.AND: continuous on [1, ?)...AND: decreasing, f'(x) < 0
Integral test (Remainder)
*Remainder*R(N) = S - S(N).
..[where S = converg. val.
& N = Nth]...AND: 0 < R(N) < [?(N to ?) f(x) dx]
(Nth) Root Test (Convergent)
Series: ? a(n)*Condition*:lim n)?|a(n)| < 1
(Nth) Root Test (Divergent)
Series: ? a(n)*Condition*:lim n)?|a(n)| > 1
(Nth) Root Test (*NON APPLICABLE*)
Series: ? a(n)*Condition*:lim n)?|a(n)| = 1
Ratio Test (Convergence)
Series: ? a(n)*Condition*:lim |___a(n+1)___| < 1......|......a (n).......|
Ratio Test (Divergence)
Series: ? a(n)*Condition*:lim |___a(n+1)___| > 1......|......a (n).......|
Ratio Test (*NON APPLICABLE*)
Series: ? a(n)*Condition*:lim |___a(n+1)___| = 1......|......a (n).......|
Direct Comparison Test (A Convergent)
Series: ? a(n)*Conditions*:0 < a(n) ? b(n)...[where b(n) > 0] ...AND: ? b(n) converges
Direct Comparison Test (B Divergent)
Series: ? a(n)*Conditions*:0 < b(n) ? a(n)...[where b(n) > 0]...AND: ? a(n) diverges
Limit Comparison Test (Convergent)
Series: ? a(n)*Conditions*:lim ___a(n)___ = L > 0..........b(n) ...[where b(n) > 0]...AND: ? b(n) converges
Limit Comparison Test (Divergent)
Series: ? a(n)*Conditions*:lim ___a(n)___ = L > 0..........b(n)...[where b(n) > 0]...AND: ? b(n) diverges
