Paul's Online Notes
Home / Calculus II / Series & Sequences / Sequences
Show Mobile Notice Show All Notes Hide All Notes
Mobile Notice
You appear to be on a device with a "narrow" screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.

### Section 10.1 : Sequences

4. Determine if the given sequence converges or diverges. If it converges what is its limit?

$\left\{ {\frac{{{{\left( { - 1} \right)}^{n - 2}}{n^2}}}{{4 + {n^3}}}} \right\}_{n = 0}^\infty$

Show All Steps Hide All Steps

Start Solution

To answer this all we need is the following limit of the sequence terms.

$\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( { - 1} \right)}^{n - 2}}{n^2}}}{{4 + {n^3}}}$

However, because of the $${\left( { - 1} \right)^{n - 2}}$$ we canâ€™t compute this limit using our knowledge of computing limits from Calculus I.

Show Step 2

Recall however, that we had a nice Fact in the notes from this section that had us computing not the limit above but instead computing the limit of the absolute value of the sequence terms.

$\mathop {\lim }\limits_{n \to \infty } \left| {\frac{{{{\left( { - 1} \right)}^{n - 2}}{n^2}}}{{4 + {n^3}}}} \right| = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^2}}}{{4 + {n^3}}} = 0$

This is a limit that we can compute because the absolute value got rid of the alternating sign, i.e. the $${\left( { - 1} \right)^{n + 2}}$$.

Show Step 3

Now, by the Fact from class we know that because the limit of the absolute value of the sequence terms was zero (and recall that to use that fact the limit MUST be zero!) we also know the following limit.

$\mathop {\lim }\limits_{n \to \infty } \frac{{{{\left( { - 1} \right)}^{n - 2}}{n^2}}}{{4 + {n^3}}} = 0$ Show Step 4

We can see that the limit of the terms existed and was a finite number and so we know that the sequence converges and its limit is zero.