Please check this convergence test (#2)

In summary, the limit comparison test was used to determine that the series $\sum_{n}\frac{1}{n.{n}^{\frac{1}{n}}}$ and $\sum_{n}\frac{1}{n}$ both diverge. Another approach is to use L'Hopital's rule to evaluate the limit of $n^{\frac{1}{n}}$, which results in a limit of 1.
  • #1
ognik
643
2
$ \sum_{n}\frac{1}{n.{n}^{\frac{1}{n}}} $

Now $\frac{1}{n}$ diverges and $\ne 0$ , so by limit comparison test:

$ \lim_{{n}\to{\infty}} \frac{n.{n}^{\frac{1}{n}}}{n} = \lim_{{n}\to{\infty}} {n}^{\frac{1}{n}} = \lim_{{n}\to{\infty}} {n}^0 = 1$ (I think the 2nd last step may be dubious?)

Therefore both series diverge

(also let me know if there is another approach, thanks)
 
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  • #2
Hey (again) ognik,

It is true that $\lim_{n\rightarrow\infty}n^{\frac{1}{n}}=1$, but not for the reason you've stated (i.e. $n^0=1$). The limit $\lim_{n}n^{\frac{1}{n}}$ is an "indeterminate form" from calculus. The trick is to write $y=n^{\frac{1}{n}}$, take a logarithm on both sides, then evaluate the limit using L'Hopital's rule to get $\lim_{n}\ln(y)=0\Longrightarrow\lim_{n}y=1.$
 
  • #3
Thanks, slowly re-learning all these tricks of the trade...

$ \lim_{{ n}\to{\infty }}ln (y) = \lim \frac{ln (n)}{n} = \lim \frac{1}{n} = 0; \therefore \lim y = e^{0} = 1$
 

FAQ: Please check this convergence test (#2)

What is a convergence test?

A convergence test is a mathematical method used to determine whether a series or sequence of numbers will converge (approach a fixed value) or diverge (not approach a fixed value) as the number of terms increases.

How do you perform a convergence test?

To perform a convergence test, you can use various methods depending on the type of series or sequence. Some common methods include the ratio test, the comparison test, and the integral test. These methods involve evaluating the behavior of the terms in the series as the number of terms increases.

Why is it important to check for convergence?

Checking for convergence is important because it allows us to determine whether a series or sequence has a definite limit or if it diverges to infinity. This information is crucial in many areas of mathematics and science, such as in calculating probabilities and analyzing the behavior of physical systems.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series or sequence that converges regardless of the order in which the terms are added. On the other hand, conditional convergence refers to a series or sequence that only converges when the terms are added in a specific order. In other words, the rearrangement of terms can change the value of a conditionally convergent series, but not an absolutely convergent one.

What happens if a convergence test fails?

If a convergence test fails, it means that the test was inconclusive and another test or method may need to be used to determine the convergence or divergence of the series or sequence. It could also indicate that the series or sequence does not have a definite limit and may diverge to infinity.

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