jsewell94 said:Homework Statement
Determine whether the following series converges or diverges:
[itex]\sum_{}^{} ( \frac{1}{3} )^{ln(n)}[/itex]
Homework Equations
N/A
The Attempt at a Solution
See attached document..
I had my Calc 2 final today, and this was our hard problem...but I don't know if my method is valid or not. Could you help me determine if it is? And if it's not, could you tell me where the flaw in my reasoning is/a better method?
Dick said:It looks ok to me. You could shorten the whole argument up a lot. [itex]3^{ln(n)}=(e^{ln(3)})^{ln(n)}=e^{ln(3) ln(n)}=(e^{ln(n)})^{ln(3)}=n^{ln(3)}[/itex]. Just use ln(3) for k.
A convergent series is a series in which the sum of its terms approaches a finite limit as the number of terms increases.
A divergent series is a series in which the sum of its terms does not approach a finite limit as the number of terms increases.
To prove that a series is convergent, you can use one of several methods, such as the comparison test, the ratio test, or the root test. These tests compare the given series to a known convergent series or use the properties of the terms to determine if the series is convergent.
To prove that a series is divergent, you can use the divergence test or the integral test. The divergence test states that if the limit of the terms of the series does not approach zero, then the series is divergent. The integral test compares the given series to an improper integral and determines if the integral is convergent or divergent.
Determining if a series is convergent or divergent is important in many areas of mathematics and science. It helps us understand the behavior of infinite sums and can be used to solve various problems, such as finding the area under a curve or predicting the growth of a population. It also allows us to determine the validity of certain mathematical equations and models.