Ratio Test vs AST

In summary, the Ratio Test and the Alternating Series Test (AST) are both methods used to determine the convergence or divergence of infinite series. The Ratio Test evaluates the limit of the ratio of consecutive terms, providing clear results for series with positive terms, while the AST specifically addresses series where terms alternate in sign, focusing on the absolute value of terms and their decreasing nature. Each test has its strengths and applicable scenarios, making them valuable tools in mathematical analysis.
  • #1
cherry
20
6
Homework Statement
Find the interval of convergence for the given power series.
Relevant Equations
N/A
Hi, I'm having difficulty understanding why the interval of convergence is (0, 18].
When I tested x=18, I got the following conclusion using the ratio test.
IMG_65D89D7F1999-1.jpeg


When I attempt using AST, the function still diverges as the lim (n -> inf) = 2^n / n ≠ 0.
What am I missing?

Thanks!
 

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  • #2
cherry said:
Homework Statement: Find the interval of convergence for the given power series.
Relevant Equations: N/A

Hi, I'm having difficulty understanding why the interval of convergence is (0, 18].
When I tested x=18, I got the following conclusion using the ratio test.
View attachment 344579

When I attempt using AST, the function still diverges as the lim (n -> inf) = 2^n / n ≠ 0.
What am I missing?

Thanks!
The series is this:

When you substitute x = 18, the numerator is not .
 
  • #3
Mark44 said:
The series is this:

When you substitute x = 18, the numerator is not .
Oh my gosh, thank you!
 
  • #4
One of the most important things to learn from mathematics is to be careful about each step. It is definitely a learning process. Mathematics is one subject where you need to get a long string of steps correct to get the right answer, and it is fairly unique in that respect. Also, when you are thinking about the hard steps, it is often the easy ones where mistakes occur. So you should make it a habit to review your work with as much attention to the easy steps as you give to the hard steps.
 
  • #5
It's a power series at the point whose radius of convergence is given by

Hence, interval of convergence contains . For we get , which converges. For we get divergence. So the interval of convergence is .
 

FAQ: Ratio Test vs AST

What is the Ratio Test?

The Ratio Test is a convergence test used in calculus to determine the absolute convergence of a series. It involves taking the limit of the absolute value of the ratio of consecutive terms in a 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.

What is AST (Alternating Series Test)?

The Alternating Series Test (AST) is a method used to determine the convergence of an alternating series, which is a series whose terms alternate in sign. The AST states that if the absolute value of the terms decreases monotonically to zero, then the series converges. This test is specifically applicable to series with alternating signs.

When should I use the Ratio Test instead of AST?

The Ratio Test is typically used when dealing with series that have positive or mixed terms, especially when the terms involve factorials, exponentials, or powers. In contrast, the AST is specifically designed for alternating series. If the series has alternating signs, the AST is more appropriate; otherwise, the Ratio Test may be more effective.

Can a series pass both the Ratio Test and the AST?

Yes, a series can pass both tests, but this is most common in series that have terms that are both alternating and meet the criteria for absolute convergence. If a series converges absolutely (as determined by the Ratio Test), it also converges conditionally, which means it will also satisfy the conditions of the AST if it is alternating.

What are the limitations of the Ratio Test and AST?

The Ratio Test has limitations in that it can be inconclusive when the limit equals 1, which means further testing is required to determine convergence. The AST is limited to alternating series and only provides information about conditional convergence; it cannot be used to determine absolute convergence. Thus, it is important to choose the appropriate test based on the nature of the series being analyzed.

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