harrietstowe said:Hey sorry to return to this late but I am still pretty stuck on this. Dick, I don't think I know how to prove that
A series absolute convergence proof is a mathematical technique used to determine if a series, which is a sequence of numbers added together, converges absolutely. Absolute convergence means that the sum of the series will always be a finite number, regardless of the order in which the terms are added. This proof is used to ensure the validity and accuracy of mathematical calculations.
A regular convergence proof only shows that a series will approach a finite limit as the number of terms increases. However, an absolute convergence proof goes a step further and shows that the series will always converge to a finite number, regardless of the order in which the terms are added. This is a stronger and more rigorous form of convergence.
Proving absolute convergence in series is important because it guarantees the accuracy and reliability of mathematical calculations. If a series is absolutely convergent, then the sum of its terms will always be the same, regardless of the order in which they are added. This allows for consistent and predictable results in mathematical computations.
Some common techniques used in series absolute convergence proofs include the comparison test, the ratio test, and the integral test. These methods involve comparing the given series to known convergent or divergent series, or using integrals to determine the convergence or divergence of the series.
Yes, there are some series that cannot be proven to be absolutely convergent. These are known as conditional convergent series, which means that the series is convergent but not absolutely convergent. These series can be proven to be convergent using traditional convergence tests, but they do not have a guaranteed finite sum regardless of the order of the terms.