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Explain the purpose of the Ratio Test.

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

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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.

Define the Ratio Test.

A test to determine the convergence or divergence of a series an\sum a_n by examining the limit of an+1an|\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.

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

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

How do you find an+1a_{n+1} given ana_n?

Replace every 'n' in the expression for ana_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 n!nn\sum \frac{n!}{n^n}, find an+1an\frac{a_{n+1}}{a_n}.

an+1an=(n+1)!(n+1)n+1nnn!=(n+1)nn(n+1)n+1=nn(n+1)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}