lukatwo said:Homework Statement
So I need to determine if the series [itex]\Sigma[/itex][itex]ln(1+e^-n)/n[/itex] converges.
Homework Equations
The Attempt at a Solution
I know it does, but cannot prove it. Wolfram says that the ratio test indicates that the series converges, but when I try to solve the limit I get that it equals 1(which is not conclusive). here
A series convergence problem is a mathematical concept that deals with determining whether an infinite sum of numbers, called a series, will result in a finite or infinite value. It involves analyzing the behavior of a series as the number of terms increases to infinity.
There are several tests that can be used to determine whether a series converges or diverges. Some common tests include the Ratio Test, the Comparison Test, and the Integral Test. These tests involve evaluating the behavior of the series and comparing it to that of known convergent or divergent series.
Absolute convergence refers to a series that converges regardless of the order in which its terms are added. On the other hand, conditional convergence refers to a series that only converges when the terms are added in a specific order. A series that is absolutely convergent is also conditionally convergent, but the reverse is not always true.
The alternating series test is a special case of the Ratio Test that is used to determine the convergence of alternating series. It states that if a series alternates between positive and negative terms and the magnitude of the terms decreases as the number of terms increases, then the series will converge.
No, a divergent series cannot be manipulated to become convergent. However, there are techniques such as partial summation and rearrangement that can alter the behavior of a series and potentially change its convergence or divergence. These techniques are often used to prove the divergence of a series.