Convergence of an infinite series

In summary, the given series is convergent and the comparison test can be used to prove it. The presence of natural log in the expression can be ignored as it does not affect the convergence. By comparing it with the convergent series 1/n^3/2, it can be shown that the original series is also convergent. Additionally, the p-series theorem states that series with p>1 are convergent.
  • #1
edoz
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Homework Statement


http://img840.imageshack.us/img840/3609/unleddn.png

note that by log(n), i really mean NATURAL log of n

Homework Equations


it's convergent, but I can't figure out which test to use

The Attempt at a Solution


there is no term to the nth power, so ratio test is useless; root test is useless too; comparison test would seem to be the best option, but I can't figure out how to compare when I have a natural log in the expression... limit comparison test takes me nowhere either.
 
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  • #2
I think that the log(n) is just there to throw you off. [tex] \log(n) < n^{1/2} [/tex] asymptotically right. Do you have a theorem showing that [tex] \sum \frac{1}{n^{3/2}} [/tex] is convergent?
 
  • #3
yes, p-series are convergen if p>1;

now i get it, I just need to compare it with the series you just shown, since with the (n+1) the original series will always be smaller than the convergent series you shown, therefore convergent too.
 

Related to Convergence of an infinite series

1. What is the definition of convergence of an infinite series?

The convergence of an infinite series refers to whether the series has a finite limit as the number of terms approaches infinity. In other words, it is the behavior of a series as the number of terms increases without bound.

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

There are various methods for determining the convergence or divergence of an infinite series, such as the ratio test, the root test, and the integral test. These tests involve analyzing the behavior of the terms of the series and can determine if the series converges to a finite limit or diverges to infinity.

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

Absolute convergence refers to a series where the absolute value of each term converges to a finite limit. On the other hand, conditional convergence refers to a series where the terms alternate in sign and the series converges to a finite limit when the absolute values of the terms are considered. In other words, absolute convergence is a stronger condition than conditional convergence.

4. Can an infinite series converge to a value other than its sum?

No, the convergence of an infinite series is defined as the limit of the series as the number of terms approaches infinity. This limit can only be equal to the sum of the series, so an infinite series cannot converge to a value other than its sum.

5. What is the role of the Cauchy criterion in determining the convergence of an infinite series?

The Cauchy criterion is a mathematical theorem that states that a series converges if and only if the partial sums of the series form a Cauchy sequence. In other words, if the terms of a series become arbitrarily close to each other as the number of terms increases, then the series is convergent. This criterion is often used in conjunction with other convergence tests to prove the convergence of an infinite series.

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