Series Question: Divergence or Converence of sum[n=1,infty][(-1)^{n+1} (n^2/(n^3+1))]

[EngStudent]

Hello! New member here! I just have a quick question on a problem I have.

. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, (-1)^{n+1}\, \left(\dfrac{n^2}{n^3\, +\, 1}\right)\)

For this problem, I used the alternating series test. Bn = n^2/(n^3+1). I showed that the limit as n approaches infinity to be 0 (2/6n = 0). Now, I just need to show that bn is decreasing. I used the first derivative test and got -x^4+2x/(x^3+1)^2. What do I do after that to show bn is decreasing or not decreasing? Thank you.

 

[pka]

Hello! New member here! I just have a quick question on a problem I have.

. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, (-1)^{n+1}\, \left(\dfrac{n^2}{n^3\, +\, 1}\right)\)

For this problem, I used the alternating series test. Bn = n^2/(n^3+1). I showed that the limit as n approaches infinity to be 0 (2/6n = 0). Now, I just need to show that bn is decreasing. I used the first derivative test and got -x^4+2x/(x^3+1)^2. What do I do after that to show bn is decreasing or not decreasing? Thank you.

Is it clear to you that if \(\displaystyle x>2\) then \(\displaystyle \dfrac{-x^4+2x}{(x^3+1)^2}<0~?\) What does a negative derivative tell us about the function?

 

[EngStudent]

Is it clear to you that if \(\displaystyle x>2\) then \(\displaystyle \dfrac{-x^4+2x}{(x^3+1)^2}<0~?\) What does a negative derivative tell us about the function?

Yes, I wasn't sure though. I concluded that the series is convergent by AST but instead of n = 1, I did n = 2. Is that right? Thanks.

 

[pka]

AST but instead of n = 1, I did n = 2. Is that right? Thanks.

It does not matter even it is not decreasing for the first \(\displaystyle 10^{10}\) terms.
Just that it is decreasing from some term on.

 

[EngStudent]

Thanks. Post can be locked now.

 

[Deleted member 4993]

Thanks. Post can be locked now.

We generally do not lock post - in case some other person have a "clarifying question" regarding the answers.

 

[lookagain]

Hello! New member here! I just have a quick question on a problem I have.

. . . . .\(\displaystyle \displaystyle \sum_{n=1}^{\infty}\, (-1)^{n+1}\, \left(\dfrac{n^2}{n^3\, +\, 1}\right)\)

For this problem, I used the alternating series test. Bn = n^2/(n^3+1).
I showed that the limit as n approaches infinity to be 0 (2/6n = 0). . . . It's 2/(6n) = 0, whether or not that that is the correct algebraic fraction.

Now, I just need to show that bn is decreasing. I used the first derivative test and got -x^4+2x/(x^3+1)^2. . . . It's (-x^4 + 2x)/(x^3 + 1)^2, with parentheses for grouping symbols.

Make sure you use grouping symbols as mentioned above.

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Instead of the derivative test, you could look at consecutive terms:

\(\displaystyle \dfrac{n^2}{n^3 \ + \ 1} \ \ \ vs. \ \ \ \dfrac{(n + 1)^2}{(n + 1)^3 \ + \ 1}\)


Divide each respective fraction's "subterms" by its numerators and compare:


\(\displaystyle \dfrac{1}{n \ + \ \dfrac{1}{n^2}} \ \ \ vs. \ \ \ \dfrac{1}{n + 1 \ + \ \dfrac{1}{(n + 1)^2}}\)


Both fractions now have a numerator of 1, but the fraction on the right involves dividing by a larger number than the fraction on the left.

Therefore, it is a decreasing series.