Convergence of ln(x)/x^1.0001 Series - Finding the Integral

In summary, the problem at hand is to determine the convergence of a series with the general term ln(x)/x^1.0001 from x=1 to x=infinity. The relevant equations to use are the series convergence tests, such as the ratio test, root test, integral test, comparison test, or limit comparison test. The integral test is likely applicable in this case, but the integral itself cannot be found. It is important to note that the logarithm of x grows slower than any positive power of x as x approaches infinity.
  • #1
Frillth
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Homework Statement



For this problem, I need to determine whether a series with the general term ln(x)/x^1.0001 will converge from x=1 to x=infinity.

Homework Equations



I'm not quite sure what the relevant equations would be for this, but I'm supposed to use one of the series convergence tests, like the ratio test, root test, integral test, comparison test, or limit comparison test.

The Attempt at a Solution



It seems like the integral test should apply to this problem, but for some reason I can't find the integral.
 
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  • #2
Well, you know that the series will diverge or converge with the associated integral, right?
Also remember that the logarithm of x grows slower to infinity than ANY positive power of x does.
 

FAQ: Convergence of ln(x)/x^1.0001 Series - Finding the Integral

What is the definition of convergence of a series?

The convergence of a series refers to the behavior of the terms in a series as the number of terms increases. A series is said to converge if the terms get closer and closer to a specific value as the number of terms increases.

What is the difference between absolute and conditional convergence?

Absolute convergence refers to a series that converges regardless of the order in which the terms are added, while conditional convergence only occurs when the terms are added in a specific order. In other words, absolute convergence guarantees convergence, while conditional convergence does not.

How do you determine if a series converges or diverges?

There are several tests that can be used to determine the convergence or divergence of a series, such as the ratio test, the comparison test, and the integral test. These tests compare the given series to a known series with known convergence behavior to determine the convergence or divergence of the given series.

What is the significance of the limit comparison test?

The limit comparison test is a useful tool for determining the convergence of a series by comparing it to a known series. By taking the limit of the ratio of the two series, we can determine if they have the same convergence behavior. If the limit is a non-zero constant, then the two series have the same convergence behavior. If the limit is zero, then the given series converges, and if the limit is infinity, then the given series diverges.

How does the alternating series test work?

The alternating series test is a special test for determining the convergence of a series that alternates between positive and negative terms. This test states that if the terms of the series decrease in absolute value and eventually approach zero, then the series converges. However, if the terms do not approach zero, or if they do not alternate in sign, then the series diverges.

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