Converge or Diverge? - # 2

[Jason76]

\(\displaystyle \lim n \rightarrow \infty \sum_{n = 1}^{\infty} (-1)^{n}\dfrac{n - 1}{n^{2} + n}\)

\(\displaystyle \lim n \rightarrow \infty \sum_{n = 1}^{\infty} (-1)^{\infty}\dfrac{(\infty) - 1}{(\infty)^{2} + (\infty)}\)

\(\displaystyle \lim n \rightarrow \infty \sum_{n = 1}^{\infty} (0)\dfrac{(\infty) - 1}{(\infty)}\) Is this reasoning correct so far?

 

[stapel]

Is this reasoning correct so far?

Dunno. Are we to assume that your subject line contains the instructions? (You included no instructions within your post.) Are you, at some point, going to tell us what your reasoning was, which caused you to post the lines you have? For instance, will you, at some point, tell us how you decided that -1, when raised to large values, becomes zero?

\(\displaystyle \lim n \rightarrow \infty \sum_{n = 1}^{\infty} (-1)^{n}\dfrac{n - 1}{n^{2} + n}\)

\(\displaystyle \lim n \rightarrow \infty \sum_{n = 1}^{\infty} (-1)^{\infty}\dfrac{(\infty) - 1}{(\infty)^{2} + (\infty)}\)

\(\displaystyle \lim n \rightarrow \infty \sum_{n = 1}^{\infty} (0)\dfrac{(\infty) - 1}{(\infty)}\)

What is the meaning of this? By what reasoning or rule are you using infinity as a numerical value? What are you trying to accomplish here? What rule or algorithm or method are you applying? What are you aiming for?

Please be complete. Because, seriously, we really can't read your mind.

 

[Jason76]

What is \(\displaystyle n^{\infty}\)

The goal of this stuff (the first part, is to see if the limit equals 0, otherwise it's divergent, unless it's indeterminate and then we can see if it equals 0 again (after dividing every term by the highest term in the denominator), if it doesn't it's divergent.) IF it comes out to 0, then a test is applied depending on the situation, to see if it's convergent of divergent.

 

[HallsofIvy]

What you've started with doesn't make sense. Either your series starts at 1 or not. If it starts at 1 then the limit has no effect.

Whether not a series starts at 0 or 1 affects the sum but not whether it converges or not. But Jason76 is referring to the fact that in order that \(\displaystyle \sum a_n\) converges, \(\displaystyle lim a_n\) must equal 0. If that is not true then the series does not converge. Of course that is useless here. The sequence obviously converges to 0 which tells us nothing about whether the series converges.