Series and Sequences

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21 Terms

1
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Geometric Series
\(\sum_{n=1}^{\infty} ar^n\) converges if \(0 \leq |r| < 1\).
2
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Geometric Series
\(\sum_{n=1}^{\infty} ar^n\) diverges if \(|r| \geq 1\).
3
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nth-term test
\(\sum_{n=1}^{\infty} a_n\) diverges if \(\lim_{n \to \infty} a_n \neq 0\).
4
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P-series
\(\sum_{n=1}^{\infty} \frac{1}{n^p}\) converges if \(p > 1\).
5
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P-series
\(\sum_{n=1}^{\infty} \frac{1}{n^p}\) diverges if \(p \leq 1\).
6
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Alternating Series
\(\sum_{n=1}^{\infty} (-1)^n a_n\) converges if 1. \(|a_{n+1}| \leq |a_n|\) and 2. \(\lim_{n \to \infty} a_n = 0\).
7
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Alternating Series
\(\sum_{n=1}^{\infty} (-1)^n a_n\) has an error bound of \(|S - S_n| < a_{n+1}\).
8
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Integral Test
\(\sum_{n=1}^{\infty} a_n\) converges if \(\int_{1}^{\infty} f(x) \, dx\) converges, where \(f(n) = a_n\) is continuous, positive, and decreasing.
9
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Integral Test
\(\sum_{n=1}^{\infty} a_n\) diverges if \(\int_{1}^{\infty} f(x) \, dx\) diverges, where \(f(n) = a_n\) is continuous, positive, and decreasing.
10
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Direct Comparison Test
\(\sum_{n=1}^{\infty} a_n\) converges if \(0 < a_n \leq b_n\) and \(\sum_{n=1}^{\infty} b_n\) converges.
11
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Direct Comparison Test
\(\sum_{n=1}^{\infty} a_n\) diverges if \(0 < b_n \leq a_n\) and \(\sum_{n=1}^{\infty} b_n\) diverges.
12
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Ratio Test
\(\sum_{n=1}^{\infty} a_n\) converges if \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| < 1\).
13
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Ratio Test
\(\sum_{n=1}^{\infty} a_n\) diverges if \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| > 1\).
14
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Ratio Test
If \(\lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = 1\), the test is inconclusive.
15
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Limit Comparison Test
\(\sum_{n=1}^{\infty} a_n\) converges if \(a_n > 0\), \(b_n > 0\), \(\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0\) (finite and positive), and \(\sum_{n=1}^{\infty} b_n\) converges.
16
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Limit Comparison Test
\(\sum_{n=1}^{\infty} a_n\) diverges if \(a_n > 0\), \(b_n > 0\), \(\lim_{n \to \infty} \frac{a_n}{b_n} = L > 0\) (finite and positive), and \(\sum_{n=1}^{\infty} b_n\) diverges.
17
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Root Test
\(\sum_{n=1}^{\infty} a_n\) converges if \(\lim_{n \to \infty} \sqrt[n]{|a_n|} < 1\).
18
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Root Test
\(\sum_{n=1}^{\infty} a_n\) diverges if \(\lim_{n \to \infty} \sqrt[n]{|a_n|} > 1\).
19
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Root Test
If \(\lim_{n \to \infty} \sqrt[n]{|a_n|} = 1\), the test is inconclusive.
20
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Conditional Convergence
\(\sum_{n=1}^{\infty} a_n\) is conditionally convergent if \(\sum_{n=1}^{\infty} a_n\) converges but \(\sum_{n=1}^{\infty} |a_n|\) diverges.
21
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Absolute Convergence
\(\sum_{n=1}^{\infty} a_n\) is absolutely convergent if \(\sum_{n=1}^{\infty} a_n\) and \(\sum_{n=1}^{\infty} |a_n|\) both converge.