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?

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

FAQ: Determining convergence of series

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.

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.

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.

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.

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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