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Typology: Cheat Sheet
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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 n→∞ an 6 = 0
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:
np Geometric Series Test:
anrn−^1 or
anrn
Direct Comparison Test:
Limit Comparison Test:
Root Test:
(an)n
Alternating Series Test:
(−1)n+1an or
(−1)nan
Ratio Test:
∑ (^) n+ 5 n Integral Test:
Remember: These are just suggestions. There are other tests which may get us to the same answer. Convergent Divergent
n=
n^2 − 1 n^2 + n
n=
n=
3 n^3
n=
(−3)n+ 23 n
n=
(−3)n+ 23 n
n=
n^2 − 1 n^2 + n
n=
tan−^1 (n) n
n
n=
n^2 − 1 n^2 + n
n=
n^2 + n
n=
n^2 − 1 n^3 + 4
n=
( (^3) n 1 + 8n
)n ∑∞
n=
(n!)n n^4 n
n=
(−1)n^
n^2 − 1 n^2 + n
n=
(−1)n^
n=
2 nn! (n + 2)!
n=
n=
e (^1) n
n^2
n=
Test for Divergence n^ lim→∞ an^6 = 0
Test for Divergence n^ lim→∞ an^6 = 0
P-Series Test
P-Series Test
Geometric Series Test
Geometric Series Test
Direct Comparison Test
Direct Comparison Test
Limit Comparison Test
Limit Comparison Test
Root Test Root Test
Alternating Series Test
Alternating Series Test
Ratio Test Ratio Test
Integral Test Integral Test
Test for Divergence Fails n^ lim→∞ an^ = 0
Test for Divergence Fails n^ lim→∞ an^ = 0