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Preliminary Test of Alternating Geometric Series

1. Homework Statement
[tex]\lim_{n \to \infty}\frac{(-1)^{n+1} \cdot n^2}{n^2+1}[/tex]

2. Homework Equations
[tex]\lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent[/tex]

3. The Attempt at a Solution
I tried L'Hopital's rule, but I could not figure out how to find the limit of that pesky [itex](-1)^{n+1}[/itex].

Edit: This is not actually a geometric series, disregard that part of the title.
 

Ray Vickson

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1. Homework Statement
[tex]\lim_{n \to \infty}\frac{(-1)^{n+1} \cdot n^2}{n^2+1}[/tex]

2. Homework Equations
[tex]\lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent[/tex]

3. The Attempt at a Solution
I tried L'Hopital's rule, but I could not figure out how to find the limit of that pesky [itex](-1)^{n+1}[/itex].

Edit: This is not actually a geometric series, disregard that part of the title.
Is there a question somewhere here? You seem to have arrived at the correct conclusion for the correct reason, so what are you unsure about?
 
Computing the derivative of an exponential function results in another exponential function, which doesn't really help when it comes to using L'Hopitals rule.
 
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[tex]\lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent[/tex]
This is a rule for series. Why is this relevant? There are no series in your post.
 

HallsofIvy

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Your post is very confusing! You title this "alternating geometric series" but appear to be asking about a "sequence" rather than a "series". Further this is not a "geometric" series or sequence. If the "problem statement" is to determine whether or not the series [itex]\sum_{n=0}^\infty \frac{(-1)^n n^2}{n^2+ 1}[/itex] converges then you should know that an alternating series, [itex]\sum_{n=0}^\infty a_n[/itex], converges if and only if the sequence [itex]a_n[/itex] converges to 0. Finally, you ask about the limit of [itex](-1)^n[/itex]. That sequence does not converge so has no limit but that is irrelevant to this problem.
 
try dividing the denominator and the numerator by n^2. you'll be left out with ones and one/infinities which go to zero then you're left with a simple equation.
 
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An alternating series is of the form ##\sum_n (-1)^na_n##. The limit in your test concerns ##a_n##, not ##(-1)^na_n##.
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The question in the first post has been answered, and the OP hasn't been back for a couple of years, so I'm closing this thread.
 

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