Calculus II Series - Things to Consider

Home , Alternating series, Alternating series test, Derivative test, Direct comparison test, Limit comparison test, Series (mathematics)

Important: This cheat sheet is not intended to be a list of guaranteed rules to follow. This intro- duces some hints and some ideas you may consider when choosing tests for convergence or divergence when evaluating a given series. It is usually a good idea to try using the Test for Divergence as a ﬁrst step when evaluating a series for convergence or divergence. If we can show that:

lim an 6= 0 n→∞ Then we can say that the series diverges without having to do any extra work.

Below are some general cases in which each test may help:

P-Series Test:

P 1 • The series be written in the form: np Geometric Series Test:

P n−1 P n • When the series can be written in the form: anr or anr Direct Comparison Test:

• When the given series, an looks like a known, or more simple, series, bn Limit Comparison Test:

• When you can see that the series looks like another convergent or divergent series, bn

• But it is hard to say whether bn > an or bn < an Root Test:

P n • When the series can be written in the form: (an) Alternating Series Test:

P n+1 P n • When the series can be written in the form: (−1) an or (−1) an Ratio Test: • Whenever we are given something involving a factorial, e.g. n!

th P n+5 • Whenever we are given something involving a constant raised to the n power, e.g. 5n Integral Test: • If the sequence is: – continuous – positive – decreasing (we can use the First Derivative Test here)

csusm.edu/stemsc xxx @csusm_stemcenter Tel: North: 760-750-4101 South: 760-750-7324 Calculus II Series - Things to Consider Remember: These are just suggestions. There are other tests which may get us to the same answer. Convergent Divergent

Test for Test for Divergence ∞ ∞ Divergence X n2 − 1 X 1 lim an 6= 0 lim an 6= 0 n→∞ n2 + n 1 n→∞ n=1 n=1

P-Series ∞ 15 ∞ (−3)n+1 P-Series Test X X Test 3n3 23n n=1 n=1

∞ ∞ Geometric X (−3)n+1 X n2 − 1 Geometric Series Test Series Test 23n n2 + n n=1 n=1

Direct Direct ∞ ∞ Comparison X tan−1(n) X n2 − 1 Comparison Test √ Test n n n2 + n n=1 n=1

Test for Test for Limit Divergence ∞ ∞ 2 Divergence Limit Comparison Fails X 1 X n − 1 Fails Comparison Test 2 3 Test lim an = 0 n + n n + 4 lim an = 0 n→∞ n=1 n=1 n→∞

∞ n ∞ n Root Test X 3n X (n!) Root Test 1 + 8n n4n n=1 n=1

∞ ∞ Alternating X n2 − 1 X 1 Alternating (−1)n (−1)n Series Test n2 + n 1 Series Test n=1 n=1

∞ ∞ X 2nn! X 1 Ratio Test Ratio Test (n + 2)! 1 n=1 n=1

∞ ∞ 1 1 X e n X Integral Test Integral Test n2 1 n=1 n=1

csusm.edu/stemsc xxx @csusm_stemcenter Tel: North: 760-750-4101 South: 760-750-7324