Preliminary Test of Alternating Geometric Series

[The-Mad-Lisper]

Homework Statement

$$\lim_{n \to \infty}\frac{(-1)^{n+1} \cdot n^2}{n^2+1}$$

Homework Equations

$$\lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent$$

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 $(-1)^{n+1}$.

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

 

[Ray Vickson]

Homework Statement

$$\lim_{n \to \infty}\frac{(-1)^{n+1} \cdot n^2}{n^2+1}$$

Homework Equations

$$\lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent$$

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 $(-1)^{n+1}$.

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?

 

[The-Mad-Lisper]

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.

 

[micromass]

$$\lim_{n \to \infty}a_n \neq 0 \rightarrow S \ is \ divergent$$

This is a rule for series. Why is this relevant? There are no series in your post.

 

[HallsofIvy]

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 $\sum_{n=0}^\infty \frac{(-1)^n n^2}{n^2+ 1}$ converges then you should know that an alternating series, $\sum_{n=0}^\infty a_n$, converges if and only if the sequence $a_n$ converges to 0. Finally, you ask about the limit of $(-1)^n$. That sequence does not converge so has no limit but that is irrelevant to this problem.

 

[Abdullah Safarini]

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.

 

[archaic]

An alternating series is of the form $\sum_n (-1)^na_n$. The limit in your test concerns $a_n$, not $(-1)^na_n$.

 

[Mark44]

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.