- #1
appletree3
- 1
- 0
prove that x/(e^x-1) + x/2 is an even function, and therefore its power series only involves even powers of x.
appletree3 said:prove that x/(e^x-1) + x/2 is an even function, and therefore its power series only involves even powers of x.
An even function is a mathematical function where the output value remains unchanged when the input value is replaced with its negative. In other words, f(x) = f(-x) for all values of x.
To prove that a function is even, we need to show that f(x) = f(-x) for all values of x. In this case, we can substitute -x in place of x in the given function and show that the resulting expression is equal to the original function.
Yes, let's take the function f(x) = x^2. To prove that it is even, we substitute -x in place of x and we get f(-x) = (-x)^2 = x^2 = f(x). This shows that the output value remains unchanged when the input value is replaced with its negative, proving that f(x) = f(-x) and therefore the function is even.
Proving a function to be even helps us to understand its symmetry and properties. It also allows us to simplify mathematical expressions and solve equations more easily. In addition, even functions are often used in real-life applications such as physics and engineering.
Yes, in addition to substituting -x in place of x, we can also use algebraic manipulation and mathematical theorems to prove that a function is even. For example, using the properties of even and odd functions, we can show that a function is even by demonstrating that it is the sum of two even functions or the product of an even and odd function.