Solving Abs Converge Issue with Ratio Test

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In summary, the book discusses the Fourier series for a function \(f\) and the use of the ratio test to determine convergence. However, the speaker has found a different solution that differs from the result obtained using the ratio test. They question if this is possible and the expert clarifies that the ratio test is inconclusive if the limit is equal to 1, and suggests the use of the limit comparison test instead.
  • #1
Dustinsfl
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The book says this isn't absolutely convergent but I keep getting it is by the ratio test. What is wrong?

The Fourier series for $f$ is $f(\theta) = 2\sum\limits_{n = 1}^{\infty}\frac{(-1)^{n + 1}}{n}\sin n\theta$.
Then
$$
-\sum\limits_{n = 1}^{\infty}\left|\frac{(-1)^{n + 1}}{n}\right|.
$$
By the ratio test, we have
\begin{alignat*}{3}
\lim_{n\to\infty}\left|\frac{(-1)^{n + 2}n}{(-1)^{n + 1}(n + 1)}\right| & = & \lim_{n\to\infty}\left|\frac{-n}{n + 1}\right|\\
& = & \lim_{n\to\infty}|-1|\frac{n}{n + 1}\\
& = & 1 < \infty
\end{alignat*}

I solved this problem in another manner but shouldn't I be able to get the same answer using the Ratio Test?
 
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  • #2
dwsmith said:
I solved this problem in another manner but shouldn't I be able to get the same answer using the Ratio Test?
The ratio test is inconclusive if the limit \(L = 1\). You are perhaps confusing this with the limit comparison test, which only requires the limit to exist and be nonzero in order to establish convergence or divergence.
 

FAQ: Solving Abs Converge Issue with Ratio Test

What is the Ratio Test method for solving an ABS converge issue?

The Ratio Test method is a mathematical tool used to determine the convergence or divergence of a series. It involves taking the ratio of consecutive terms in a series and analyzing its behavior as the number of terms increases. This method is commonly used to solve convergence issues in ABS series.

How does the Ratio Test determine convergence or divergence?

The Ratio Test states that if the limit of the ratio of consecutive terms in a series is less than 1, then the series converges. If the limit is greater than 1, then the series diverges. If the limit is equal to 1, then the test is inconclusive and other methods may need to be used.

Can the Ratio Test be used to solve any ABS converge issue?

No, the Ratio Test can only be used for series that follow certain criteria, such as having positive terms and decreasing in magnitude. If the series does not meet these criteria, then other methods, such as the Root Test or the Integral Test, may need to be used.

What should I do if the limit in the Ratio Test is inconclusive?

If the limit in the Ratio Test is inconclusive, it means that the test cannot determine whether the series converges or diverges. In this case, you may need to use other methods or perform further analysis to determine the convergence or divergence of the series.

Is the Ratio Test always reliable for solving ABS converge issues?

No, the Ratio Test is not always reliable. There are some series, such as alternating series, that may satisfy the criteria for the Ratio Test but still diverge. It is important to always check the conditions and consider other methods when using the Ratio Test to solve ABS convergence issues.

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