Quotient Criterion

Definition

Quotient Criterion

The quotient criterion is a convergence criterion for infinite series $\sum_{n \geq 0} a_{n}$ .

$\exists q < 1 : (\frac{a _{n + 1}}{a _{n}}) \leq q$ for almost all $n$ $⟹ \sum_{n \geq 0} a_{n}$ absolute convergent

$\frac{a _{n + 1}}{a _{n}} \geq 1$ for almost all $n$ $⟹ \sum_{n \geq 0} a_{n}$ divergent

Limit (superior / inferior) form:

$lim sup_{n \to \infty} \frac{a _{n + 1}}{a _{n}} < 1 ⟹ > \sum_{n \geq 0} a_{n}$ convergent

$lim in f_{n \to \infty} \frac{a _{n + 1}}{a _{n}} > 1 ⟹ \sum_{n \geq 0} a_{n}$ divergent

Example

Let $(a_{n})_{n \geq 0}$ be a sequence:

a_{n} \frac{a _{n + 1}}{a _{n}} ⟹ n \to \infty lim sup \frac{a _{n + 1}}{a _{n}} = \frac{n}{2 ^{n}} = \frac{( \frac{n + 1}{2 ^{n + 1}} )}{( \frac{n}{2 ^{n}} )} = \frac{n + 1}{2 n} = \frac{n}{2 n} = \frac{1 + \frac{1}{n}}{2} ⟶ \frac{1}{2} = \frac{1}{2}

According to the quotient criterion:

n \to \infty lim sup \frac{a _{n + 1}}{a _{n}} = \frac{1}{2} < 1 ⟹ n \geq 0 \sum a_{n} convergent

Lukas' Notes

Quotient Criterion

Definition

Example

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