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Math 142 Series
Mnemonic: PARTING CC
P-series, Alternating Series Test, Ratio and Root Tests, Telescoping Series, Integral Test, N-th Term Test, Geometric Series Test Comparison Test, Limit Comparison Test
p-Series
Form:
โ^ โ
n=
np
Converges: If p > 1. Diverges: If p โค 1.
Remember to look for some constant multiple of this series also. (Like an = (^) n^25 )
Alternating Series Test
Form:
โ^ โ
n=
(โ1)nun or
โ^ โ
n=
(โ1)n+1un
Converges: If (a) un+1 โค un for all n and (b) lim nโโ un = 0.
This test only applies to alternating series. Also, note that we only need that {un} is eventually a decreasing sequence since a finite number of terms does not affect convergence.
Ratio Test
Suppose we have the series
an. Define L = lim nโโ
an+ an
(1) If L < 1 the series is absolutely convergent (and hence convergent). (2) If L > 1 the series is divergent. (3) If L = 1 the series may be divergent, conditionally convergent, or absolutely convergent.
Use this test when the series contains factorials or other products (including a constant raised to the nth power).
Root Test
Suppose we have the series
an. Define L = (^) nlimโโ^ n
|an| = (^) nlimโโ |an|^1 /n.
(1) If L < 1 the series is absolutely convergent (and hence convergent). (2) If L > 1 the series is divergent. (3) If L = 1 the series may be divergent, conditionally convergent, or absolutely convergent.
Try this test when an = (bn)n.
Telescoping Series
A telescoping series is a series whose partial sums SN eventually only have a fixed number of terms after cancellation. You can then evaluate the limit lim N โโ SN to determine whether the series converges or diverges.
(Partial fraction decomposition may be necessary.)
1
2
Integral Test
Suppose that f (x) is a continuous, positive, decreasing function on the interval [k, โ) and that f (n) = an then
(1) If
k
f (x) dx is convergent so is
โ^ โ
n=k
an.
(2) If
k
f (x) dx is divergent so is
โ^ โ
n=k
an.
This test only applies to series that have positive terms. Try this test when f (x) is easy to integrate.
N -th Term Test for Divergence
If lim nโโ an 6 = 0 then
an diverges.
Look for the same degree in the numerator and denominator. Remember: If lim nโโ an = 0 then we know nothing.
Geometric Series
Form:
โ^ โ
n=
arnโ^1
Converges: If |r| < 1. Diverges: If |r| โฅ 1.
Some algebraic manipulation is often required to get a geometric series into the correct form. (Remember r is the common ratio: r = an a+1n .)
Comparison Test
Suppose that
an and
bn are series with positive terms:
(1) If
bn is convergent and an โค bn for all n, then
an is also convergent. (2) If
bn is divergent and an โฅ bn for all n, then
an is also divergent.
Use this test if the given series is similar to a p-series or geometric series. Remember, the terms of the series being tested must be smaller than a convergent series or larger than a divergent series. This test is often a last resort; other tests are often easier to apply.
Limit Comparison Test
Suppose that
an and
bn are series with positive terms:
If lim nโโ
an bn
= C, where C > 0 is a finite number, then either both series converge or both series diverge.
Try this test if an is a rational expression involving only polynomials or polynomials under radicals. This test is easier to apply than the Comparison Test.
