Ratio Test
Definition
The ratio test is used to determine whether an infinite series converges absolutely, converges conditionally, or diverges by examining the limit of the ratio of consecutive terms.
Analogy
Think of the ratio test as a way to determine if a population is growing or declining. If the ratio of the number of individuals in one generation to the number in the previous generation approaches a certain value (less than 1), then the population will eventually decline and converge. But if that ratio exceeds a certain value (greater than 1), then the population will grow indefinitely and diverge.
Related terms
Absolute Convergence: A type of convergence where both positive and negative terms in a series converge individually.
Conditional Convergence: A type of convergence where rearranging the order of terms in a convergent series can change its sum.
Divergence Test: A method used to determine whether an infinite series diverges by checking if its terms do not approach zero.
"Ratio Test" appears in:
Study guides (2)
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Practice Questions (6)
What does the Ratio Test state?
Determine if the following series converge or diverge using the Ratio Test. ``` ∑ (2^n)/(n+1)! ```
Determine if the following series converge or diverge using the Ratio Test. ``` ∑ (2n)! / (3n + 10) ```
Determine if the following series converge or diverge using the Ratio Test. ``` ∑ ((4n-1)!) / (7n)! ```
Which kinds of series is the ratio test useful for?
Determine if the following series converge or diverge using the Ratio Test. ``` ∑ ((7n-5)!) / (12n)! ```
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