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Level 628

Divergence & Convergence of Series


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Nth Term Test
If Limit as K approaches infinity, then the Series of K Diverges.
Alternating Series Test
If An is positive, the series ∑(-1)An converges if & only if..
Geometric Series Test
general term = a₁r^n, converges if -1 < r < 1
P-Series Test
general term = 1/n^p, converges if p > 1
Integral Test
take integral and evaluate, if it goes to a number it is convergent, if there is ∞in it, its divergent
Root Test
if limit as n nears infinity of the nth root of a sub n is less than 1, the series CONVERGES; and if the limit is greater than 1 or infinity, the series DIVERGES
Comparison Test
after comparing remember to take the limit as n→∞ of an/bn and for convergence it must lie on the interval 0<L<∞
Ratio Test
limit as n→∞ of an+1/an
nth-term
Diverges: lim n→∞ an ≠ 0
geometric series
sum of the terms of an arithmetic sequence
telescoping series
Converges: lim→∞ bn = L
P-Series
Sum where 1/(n^p) converges if and only if p > 1
integral
(works if f(x) is continuous, positive, decreasing)
root
A root of a polynomial in one variable is a value of the variable for which the polynomial is equal 0.
ratio
A comparison of two quantities, also know as fancy word for fraction.
0<an≤bn
direct comparison
limit comparison
Series converges if lim (as n approaches infinity) a{n}/ b{n}= L>0 and