Prove that x/(e^x-1) + x/2 is an even function, and therefore its

• appletree3
In summary, an even function is a mathematical function where the output value remains unchanged when the input value is replaced with its negative. To prove that a function is even, we can substitute -x in place of x and show that the resulting expression is equal to the original function. This helps in understanding symmetry and simplifying mathematical problems. Other methods to prove a function is even include using algebraic manipulation and mathematical theorems.
appletree3
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.

well to show that it is even is not difficult at all, i tried it myself and it works nicely!

all you need to do is show that f(-x)=f(x), this method works, although there might be others as well.

1. What is an even function?

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.

2. How can I prove that x/(e^x-1) + x/2 is an even function?

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.

3. Can you provide an example of proving an even 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.

4. How does proving a function to be even help in solving mathematical problems?

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.

5. Are there any other methods to prove that a function is even?

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.

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