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Positive Term Series

Integral Test

Let \( \sum a_n \) be a positive term series, and let \( a_n = f(n) \) such that \( f(n) \) decreases as \( n \) increases. Then \( \sum a_n \) converges or diverges if \( \int_1^\infty f(x) dx \) is finite or infinite respectively.

p-Series Test

Let \( \sum a_n \) be a positive term series given by \(a_n = \frac1{n^p} \). Then, \( \sum a_n \) is convergent if \( p \gt 1 \), and divergent if \( p \leq 1 \).

Comparison Test

Let \(\sum a_n \) be a positive term series, then:

1. \( \sum a_n \) is convergent if \( \sum b_n \) is another convergent series with \( a_n \leq b_n \).
2. \( \sum a_n \) is divergent if \( \sum d_n \) is another divergent series with \( a_n \geq d_n \).

Limit Comparison Test

Let \( \sum a_n \) and \(\sum b_n \) be two positive term series.

1. If \( \lim_{n \rightarrow \infty} \frac{a_n}{b_n} \) is a finite and non-zero positive quantity, then \( \sum a_n \) and \(\sum b_n\) will converge and diverge together.
2. If \( \lim_{n \rightarrow \infty} \frac{a_n}{b_n} = 0 \) and \(\sum b_n \) is convergent, then \( \sum a_n \) is also convergent.
3. If \(\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = \infty \) and \( \sum b_n \) is divergent, then \(\ sum a_n \) is also divergent.

D’Alembert’s Ratio Test / Ratio Test

Let \( \sum a_n \) be a positive term series, and let \( \lim_{n \rightarrow \infty} \frac{a_{n+1}}{a_n} = r \).

1. The series is convergent if \( r \lt 1 \).
2. The series is divergent if \( r \gt 1 \), or if \(r\) is infinite.
3. The test fails if \(r=1\).

Cauchy’s Root Test / Root Test

Let \( \sum a_n \) be a positive term series, and \( \lim_{n \rightarrow \infty} (a_n)^{\frac1n} = r \).

1. The series is convergent if \( r \lt 1 \).
2. The series is divergent if \( r \gt 1 \).
3. The test fails if \( r = 1 \).

Raabe’s Test

Let \( \sum a_n \) be a positive term series, and \( \lim_{n \rightarrow \infty} n \left( \frac{a_n}{a_{n+1}}-1 \right) = k\).

1. The series is convergent if \( k \gt 1 \).
2. The series is divergent if \( k \lt 1 \).
3. The test fails if \( k = 1 \).

Logarithmic Test

Let \( \sum a_n \) be a positive term series, and \( \lim_{n \rightarrow \infty} \log\left( \frac{a_n}{a_{n+1}} \right) = k\).

1. The series is convergent if \( k \gt 1 \).
2. The series is divergent if \( k \lt 1 \).
3. The test fails if \( k = 1 \).

Alternating Series

Leibniz’s Test

If the series \( \sum (-1)^n a_n \) is an alternating series, then the series is convergent if:

1. Each term is numerically lesser than the preceeding term. ( \( |a_{n+1}| \lt |a_n| \) )
2. \( \lim_{n \rightarrow \infty} a_n = 0 \).