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

∞ X 1 Form: np n=1 Converges: If p > 1. Diverges: If p ≤ 1.

2 Remember to look for some constant multiple of this series also. (Like an = n5 )

Alternating Series Test

∞ ∞ X n X n+1 Form: (−1) un or (−1) un n=1 n=1

Converges: If (a) un+1 ≤ un for all n and (b) lim un = 0. n→∞

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 P an+1 Suppose we have the series an. Define L = lim . n→∞ 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 P pn 1/n Suppose we have the series an. Define L = lim |an| = lim |an| . n→∞ 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. n Try this test when an = (bn) .

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 SN to determine whether the series converges or diverges. N→∞ (Partial fraction decomposition may be necessary.) 1 2

Integral Test

Suppose that f(x) is a continuous, positive, decreasing function on the interval [k, ∞) and that f(n) = an then

∞ Z ∞ X (1) If f(x) dx is convergent so is an. k n=k ∞ Z ∞ X (2) If f(x) dx is divergent so is an. k n=k 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

P If lim an 6= 0 then an diverges. n→∞ Look for the same degree in the numerator and denominator. Remember: If lim an = 0 then we know nothing. n→∞

Geometric Series

∞ X Form: arn−1 n=1 Converges: If |r| < 1. Diverges: If |r| ≥ 1.

Some algebraic manipulation is often required to get a geometric series into the correct form. (Remember r is the common ratio: r = an+1 .) an

Comparison Test

P P Suppose that an and bn are series with positive terms:

P P (1) If bn is convergent and an ≤ bn for all n, then an is also convergent. P P (2) If bn is divergent and an ≥ 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

P P Suppose that an and bn are series with positive terms: a If lim n = C, where C > 0 is a finite number, then either both series converge or both series diverge. n→∞ bn

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.