Define the Ratio Test.

A test to determine the convergence or divergence of a series $\sum a_n$ by examining the limit of $|\frac{a_{n+1}}{a_n}|$ .

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Define the Ratio Test.

A test to determine the convergence or divergence of a series $\sum a_n$ by examining the limit of $|\frac{a_{n+1}}{a_n}|$ .

What does 'indeterminate' mean in the context of the Ratio Test?

It means the Ratio Test is inconclusive, and another test is needed to determine convergence or divergence.

What type of series is the Ratio Test most useful for?

Series involving exponentials or factorials.

Explain the purpose of the Ratio Test.

To determine whether an infinite series converges or diverges by analyzing the ratio of consecutive terms.

What does the Ratio Test tell us when L = 0?

The series converges absolutely.

Why is the absolute value important in the Ratio Test?

It handles series with alternating signs, ensuring the limit reflects the magnitude of term changes.

Steps to apply the Ratio Test to $\sum a_n$ ?

Find $a_{n+1}$ . 2. Compute $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$ . 3. If $L < 1$ , converges; $L > 1$ , diverges; $L = 1$ , inconclusive.

How do you find $a_{n+1}$ given $a_n$ ?

Replace every 'n' in the expression for $a_n$ with '(n+1)'.

What should you do if the Ratio Test is indeterminate?

Use a different convergence/divergence test (e.g., Integral Test, Comparison Test).

Given $\sum \frac{n!}{n^n}$ , find $\frac{a_{n+1}}{a_n}$ .

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

All Flashcards

Define the Ratio Test.

A test to determine the convergence or divergence of a series $\sum a_n$ by examining the limit of $|\frac{a_{n+1}}{a_n}|$ .

What does 'indeterminate' mean in the context of the Ratio Test?

It means the Ratio Test is inconclusive, and another test is needed to determine convergence or divergence.

What type of series is the Ratio Test most useful for?

Series involving exponentials or factorials.

Explain the purpose of the Ratio Test.

To determine whether an infinite series converges or diverges by analyzing the ratio of consecutive terms.

What does the Ratio Test tell us when L = 0?

The series converges absolutely.

Why is the absolute value important in the Ratio Test?

It handles series with alternating signs, ensuring the limit reflects the magnitude of term changes.

Steps to apply the Ratio Test to $\sum a_n$ ?

Find $a_{n+1}$ . 2. Compute $\lim_{n \to \infty} |\frac{a_{n+1}}{a_n}| = L$ . 3. If $L < 1$ , converges; $L > 1$ , diverges; $L = 1$ , inconclusive.

How do you find $a_{n+1}$ given $a_n$ ?

Replace every 'n' in the expression for $a_n$ with '(n+1)'.

What should you do if the Ratio Test is indeterminate?

Use a different convergence/divergence test (e.g., Integral Test, Comparison Test).

Given $\sum \frac{n!}{n^n}$ , find $\frac{a_{n+1}}{a_n}$ .

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