Determining convergence of series

In summary, for the series n=1 to infinity for (-1)^n / (n^3)-ln(n), the expert determined that it converges absolutely using the limit comparison test with the comparison function c(x) = 1/n^3, as 1/n^3 is known to converge by the p-series test. The limit comparison test was used to compare the given series with the comparison series, and the limit approached 1 as n approached infinity, indicating absolute convergence. The expert also clarified that the term n^3 is dominant over ln(n) near infinity, providing further support for the absolute convergence of the series.
  • #1
calcboi
16
0
I have a question on which test to use for series n=1 to infinity for (-1)^n / (n^3)-ln(n) in order to determine convergence/divergence. I am pretty sure I determined it converges through the Alternating Series Test(correct me if I'm wrong) but I am not sure whether it is conditional or absolute. I tried the Direct Comparison Test but it was inconclusive, and I am stuck now on what to do. I also tried Limit Comparison but the limit goes to infinity so it is also inconclusive. Can you please help?
 
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  • #2
Yes you are correct.

near infinity the term n^3 is dominant over ln (n).
 
  • #3
BTW, I used the comparison c(x) = 1/n^3 for DCT and LCT since we know 1/n^3 converges by p-series.
 
  • #4
calcboi said:
BTW, I used the comparison c(x) = 1/n^3 for DCT and LCT since we know 1/n^3 converges by p-series.

Right, $\displaystyle\lim_{n\to \infty}\left(\frac{1}{n^3-\log n}:\frac{1}{n^3}\right)=\ldots =1\neq 0$, so the series is absolutely convergent.
 
  • #5
What test did you use to determine absolute convergence? Or was that just analyzing end behavior?
 
  • #8
calcboi said:
When I tried the Limit Comparison Test, I got infinity as n approaches infinity. How did you get 1?

$\displaystyle\lim_{n\to \infty}\left(\frac{1}{n^3-\log n}:\frac{1}{n^3}\right)=\lim_{n\to \infty}\frac{n^3}{n^3-\log n}=\lim_{n\to \infty}\frac{1}{1-(\log n/n^3)}=\frac{1}{1-0}=1$
 

Related to Determining convergence of series

1. What is the definition of convergence in a series?

The concept of convergence in a series refers to the behavior of the terms in the series as the number of terms approaches infinity. If the terms in the series approach a finite limit, the series is said to converge. If the terms do not approach a finite limit, the series is said to diverge.

2. How do you determine if a series is convergent or divergent?

To determine if a series is convergent or divergent, you can use several tests such as the comparison test, ratio test, or integral test. These tests compare the given series to a known convergent or divergent series to determine its behavior.

3. What is the difference between absolute and conditional convergence?

Absolute convergence refers to a convergent series where the terms are all positive. Conditional convergence refers to a convergent series where the terms alternate between positive and negative values. In other words, absolute convergence does not depend on the order of the terms, while conditional convergence does.

4. Can a series be both convergent and divergent?

No, a series cannot be both convergent and divergent. A series can either approach a finite limit, making it convergent, or not approach a finite limit, making it divergent. It cannot exhibit both behaviors simultaneously.

5. How does the rate of convergence affect the overall convergence of a series?

The rate of convergence refers to how quickly the terms in a series approach the finite limit. A series with a faster rate of convergence will converge more quickly than a series with a slower rate. However, both series will still converge as long as the terms approach a finite limit, regardless of the rate of convergence.

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