the_kid said:Homework Statement
I'm trying to compute the sum of the following series:
S=1+[itex]\frac{1}{4}[/itex]-[itex]\frac{1}{16}[/itex]-[itex]\frac{1}{64}[/itex]+[itex]\frac{1}{256}[/itex]
Homework Equations
The Attempt at a Solution
I'm not really sure how to begin this one. I know it probably involves Riemann's Rearrangement Theorem since this series is absolutely convergent.
The Riemann Rearrangement Theorem is a mathematical theorem that states that given an infinite series of real numbers, the terms of the series can be rearranged in any way and still converge to the same limit, as long as the series is conditionally convergent.
A series is said to be conditionally convergent if it converges, but not absolutely. This means that while the series itself converges, the absolute value of the terms in the series do not converge.
No, the Riemann Rearrangement Theorem can only be applied to conditionally convergent series. For absolutely convergent series, the terms must be rearranged in a specific order in order to maintain the same limit.
The Riemann Rearrangement Theorem was first proven by German mathematician Bernhard Riemann in the mid-19th century. His proof involved using a technique called the Dirichlet test, which is commonly used to test for the convergence of infinite series.
The Riemann Rearrangement Theorem has applications in various fields such as physics, engineering, and economics. It is used to analyze and model complex systems that involve infinite series, such as fluid dynamics and signal processing. It is also used in financial forecasting and risk management.