To prove |ln 2|, you need to use the definition of absolute value and the properties of logarithms. First, write ln 2 as ln(2) and then use the property ln(a) = b if and only if e^b = a. This will help you to simplify the expression and prove that |ln 2| = ln 2.
To prove |sin x|, you can use the properties of absolute value and trigonometric identities. One approach is to use the identity |sin x| = √(sin^2x). Then, you can use the Pythagorean identity sin^2x + cos^2x = 1 to simplify the expression and prove that |sin x| = sin x.
Yes, there are a few tips for proving series. First, make sure to clearly state the series you are trying to prove and the value it is equal to. Then, use mathematical induction or other methods to show that the series is convergent. Additionally, it can be helpful to use known series and their properties to simplify the series you are trying to prove.
To use limits to prove series, you can use the definition of a limit and properties of limits. For example, you can use the limit comparison test or the ratio test to show that the limit of the series is equal to a known value, which proves the series is convergent.
Proving series is important because it helps to determine the convergence or divergence of a series, which can have real-world applications in fields such as physics, engineering, and finance. It also helps to build a better understanding of mathematical concepts and their properties.