If k is an integer, what is cos(πk) ?Chaoticoli said:Homework Statement
limit as k--> infinity ABS VALUE((cos(pi*k+pi))/(cos(k*pi)))
Homework Equations
The Attempt at a Solution
Can someone prove to me why this limit is equal to 1? I have tried several other sources and I have not had any luck.
Chaoticoli said:Homework Statement
limit as k--> infinity ABS VALUE((cos(pi*k+pi))/(cos(k*pi)))
I have tried several other sources and I have not had any luck.
Chaoticoli said:It must alternate between -1 and 1.
Absolute convergence refers to the convergence of a series without considering the signs of its terms. In other words, the series will converge regardless of whether the terms are positive or negative. This is important in determining the limit of a series, as it allows us to disregard the signs and focus on the magnitude of the terms.
To prove that the limit of an absolutely convergent series is equal to 1, we use the definition of absolute convergence, which states that a series is absolutely convergent if the sum of the absolute values of its terms is finite. Then, we use the definition of a limit to show that the limit of the series is equal to 1 as the number of terms approaches infinity.
Yes, a series can be absolutely convergent but have a limit other than 1. The absolute convergence of a series only guarantees that the series will converge, but it does not determine the value of the limit. It is possible for a series to have a limit of 1 but not be absolutely convergent, or to be absolutely convergent but have a limit other than 1.
One common example is the geometric series, defined as a series where each term is a constant multiple of the previous term. As long as the absolute value of the constant is less than 1, the series will be absolutely convergent with a limit of 1. Another example is the harmonic series, where each term is the reciprocal of a natural number. This series is also absolutely convergent with a limit of 1.
Proving absolute convergence and a limit of 1 is important in mathematics because it helps us understand the behavior of series. It allows us to determine if a series will converge or diverge, and if it does converge, what its limit will be. This is essential in many areas of mathematics, including calculus, analysis, and number theory.