# Raabe's test says Legendre functions always converge?

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In summary, Raabe's test is a convergence test used in mathematics to determine the convergence or divergence of a series. It can be applied to Legendre functions to determine their convergence. It is important to know if Legendre functions converge as it indicates whether the series will approach a specific value or diverge to infinity. However, Raabe's test is not always a guarantee for convergence and further analysis may be needed. This test can also be applied to other types of series with non-negative terms. Other tests such as the ratio test and root test can also be used to determine the convergence of Legendre functions. It is recommended to use multiple tests to confirm the convergence or divergence of a series.
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The Legendre functions are the solutions to the Legendre differential equation. They are given as a power series by the recursive formula (link [1] given below):

##\begin{align}y(x)=\sum_{n=0}^\infty a_n x^n\end{align}##

##\begin{align}a_{n+2}=-\frac{(l+n+1)(l-n)}{(n+1)(n+2)}a_n\end{align}##

If ##l## is a non-negative integer, The Legendre functions reduce to the Legendre polynomials. Otherwise, they are divergent (for ##x\in[-1,1]##).

However, I did Raabe's test and found that they are convergent for all values of ##l##, a wrong result.

My workings:

##\begin{align}y(1)&=\sum_{n=0}^\infty a_n\,1^n\\
&=\sum_{n=0,2,4...}^\infty a_n+\sum_{n=1,3,5...}^\infty a_n\end{align}##

Let ##\,y_1=\sum_{n=0,2,4...}^\infty a_n\,\,\,\,\,\,\,\,## and ##\,\,\,\,\,\,\,\,y_2=\sum_{n=1,3,5...}^\infty a_n##

Applying Raabe's test to ##\,y_1## (note that the ##\,a_{n+2}## term in (2) becomes the ##\,a_{n+1}## term in ##y_1##),

##\begin{align}R&=\displaystyle\lim_{n\rightarrow +\infty}n\big(\big|{\frac{a_n}{a_{n+1}}}\big|-1\big)\\
&=\displaystyle\lim_{n\rightarrow +\infty}n\big[\frac{(n+1)(n+2)}{(l+n+1)(n-l)}-1\big]\\
&=\frac{2n^2+\big(l(l+1)+2\big)n}{n^2+n-l(l+1)}\\
&=2\end{align}##

##R>1##. Thus, ##\,y_1## is absolutely convergent.

Since both ##\,y_1## and ##\,y_2## follows the same recursive formula (2), ##\,y_2## is also absolutely convergent. So ##\,y(1)## is absolutely convergent. Since ##\,y(1)## is absolutely convergent, ##\,y(-1)## is convergent.

But ##\,y(1)## or ##\,y(-1)## must diverge. Where's my mistake?

Background information:

http://mathworld.wolfram.com/LegendreDifferentialEquation.html

http://en.wikipedia.org/wiki/Ratio_test

Thank you for your post. After reviewing your calculations, I believe there may be a mistake in your application of Raabe's test. In step 4, you have written that R>1, which would imply that the series is divergent. However, the test states that if R>1, the series is inconclusive and further tests must be done. If R=1, the series is inconclusive and if R<1 the series is convergent. Therefore, your result does not necessarily mean that the series is absolutely convergent. I recommend double checking your calculations and applying the test again. Additionally, it may be helpful to try using a different convergence test to verify your result.

I hope this helps clarify the issue. Please let me know if you have any further questions or concerns.

## 1. What is Raabe's test and how does it relate to Legendre functions?

Raabe's test is a convergence test used in mathematics to determine the convergence or divergence of a series. It is specifically used to test the convergence of series with non-negative terms. Legendre functions are a type of special functions used in mathematical physics, and Raabe's test can be applied to determine their convergence.

## 2. Why is it important to know if Legendre functions converge?

Knowing whether or not Legendre functions converge is important because it indicates whether the series will approach a specific value or if it will diverge to infinity. This information is crucial in many mathematical and scientific applications, such as in solving differential equations and modeling physical phenomena.

## 3. Does Raabe's test always guarantee convergence for Legendre functions?

No, Raabe's test is not a foolproof method for determining convergence. It is only a sufficient condition, meaning that if the test shows convergence, then the series will definitely converge. However, if the test shows divergence, further analysis is needed to determine the actual convergence or divergence of the series.

## 4. Can Raabe's test be applied to other types of series?

Yes, Raabe's test can be applied to other types of series with non-negative terms, not just Legendre functions. It is a general test for determining convergence and can be used in various mathematical and scientific contexts.

## 5. Are there any other tests that can be used to determine the convergence of Legendre functions?

Yes, there are other convergence tests that can be applied to Legendre functions, such as the ratio test and the root test. These tests may provide different results, so it is important to use multiple tests to confirm the convergence or divergence of a series.

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