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Ratio Test

Definition

Definition of Ratio Test

Let $\sum_{n=1}^{\infty} a_n$ be a positive-term series with $a_n > 0$ . If:

$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho$

Then:

符号说明

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$\rho$	Greek letter	Rho	Represents the limit value in series convergence tests
$\sum$	Greek letter	Sigma	Summation symbol, representing series
$\infty$	Mathematical symbol	Infinity	Represents infinite series, infinite number of terms
$\lim$	Mathematical symbol	Limit	Represents limit of sequence or function

Ratio Test Formula

$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \rho$

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{n!}{n^n}$ .

Solution: $a_n = \frac{n!}{n^n}$

$\frac{a_{n+1}}{a_n} = \frac{(n+1)!}{(n+1)^{n+1}} \cdot \frac{n^n}{n!} = \frac{n^n}{(n+1)^n} = \left(\frac{n}{n+1}\right)^n$

$\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \left(\frac{n}{n+1}\right)^n = \frac{1}{e} < 1$

Therefore, the series converges.

Determine the convergence of the series $\sum_{n=1}^{\infty} \frac{n}{2^n}$ .

Reference Answer(2 个标签)

series convergenceratio test

Problem-solving approach: Use the ratio test to compute the limit of the ratio between consecutive terms.

Detailed steps:

Let $a_n = \frac{n}{2^n}$
Calculate the ratio: $\frac{a_{n+1}}{a_n} = \frac{n+1}{2^{n+1}} \cdot \frac{2^n}{n} = \frac{n+1}{2n}$
Find the limit: $\lim_{n \to \infty} \frac{a_{n+1}}{a_n} = \lim_{n \to \infty} \frac{n+1}{2n} = \frac{1}{2} < 1$
Determine convergence: Ratio is less than 1, so the series converges

Answer: The series converges (convergence).

Symbol	Type	Pronunciation/Explanation	Meaning in This Article
$e$	Mathematical symbol	Euler’s number	Base of the natural logarithm, approximately 2.71828

Chinese Term	English Term	IPA Pronunciation	Explanation
比值判别法	ratio test	/ˈreɪʃiəʊ test/	Method to determine series convergence using the ratio of consecutive terms
达朗贝尔判别法	d’Alembert’s test	/dælˈæmbəts test/	Another name for the ratio test
正项级数	positive series	/ˈpɒzətɪv ˈsɪəriːz/	Series where all terms are non-negative
收敛	convergence	/kənˈvɜːdʒəns/	Sequence of partial sums has a finite limit
发散	divergence	/daɪˈvɜːdʒəns/	Sequence of partial sums has no finite limit