Ratio test proof

In summary, the Ratio Test is a method used to determine the convergence or divergence of infinite series. The test involves calculating the limit of the absolute value of the ratio of consecutive terms in the series as the index approaches infinity. If the limit is less than 1, the series converges; if greater than 1, it diverges; and if it equals 1, the test is inconclusive. The proof relies on the comparison of the terms' growth rates and utilizes the properties of limits to establish the criteria for convergence.
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Homework Statement
Please see below
Relevant Equations
Please see below
For (a) and (b),
1716794684518.png

Does someone please know how to prove this? I don't have any ideas where to start.

Thanks!
 
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a) b) is standard theory.

Relevant examples also included in the link above.
 
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A good starting point is to compare it to a geometric series. For example if ##c=1/3## can you think of a series that converges whose terms are eventually guaranteed to be larger than the ##x_n##?
 
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If the ratio is > 1, then compare to adding a nonzero number to itself " infinitely often", show it will eventually surpass any finite value.
 
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FAQ: Ratio test proof

What is the Ratio Test in calculus?

The Ratio Test is a method used to determine the convergence or divergence of an infinite series. It involves examining the limit of the absolute value of the ratio of consecutive terms in the series as the index approaches infinity. If this limit exists, it can help classify the series as absolutely convergent, divergent, or inconclusive.

How do you apply the Ratio Test?

To apply the Ratio Test, follow these steps: 1) Identify the terms of the series, typically denoted as \( a_n \). 2) Compute the limit \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| \). 3) Analyze the value of \( L \): If \( L < 1 \), the series converges absolutely; if \( L > 1 \) or \( L = \infty \), the series diverges; if \( L = 1 \), the test is inconclusive, and other methods must be used.

What conditions must be met for the Ratio Test to be applicable?

The Ratio Test can be applied to series of non-negative terms or when the terms can be made non-negative by taking absolute values. It is particularly useful for series involving factorials, exponentials, or power series. However, it may not be applicable if the limit does not exist or if the terms do not approach zero.

Can you provide an example of the Ratio Test in action?

Sure! Consider the series \( \sum_{n=1}^{\infty} \frac{n!}{n^n} \). To apply the Ratio Test, calculate \( L = \lim_{n \to \infty} \left| \frac{a_{n+1}}{a_n} \right| = \lim_{n \to \infty} \frac{(n+1)!/(n+1)^{n+1}}{n!/n^n} \). Simplifying this gives \( L = \lim_{n \to \infty} \frac{(n+1) n^n}{(n+1)^{n+1}} = \lim_{n \to \infty} \frac{n^n}{(n+1)^n} = \lim_{n \to \infty} \left( \frac{n}{n+1} \right)^n = \frac{1}{e} < 1 \). Therefore, the series converges absolutely.

What happens if the Ratio Test is inconclusive?

If the Ratio Test yields \( L = 1 \), the test is inconclusive, and you

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