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

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SUMMARY

The function f(x) = x/(e^x - 1) + x/2 is proven to be an even function, as demonstrated by the equality f(-x) = f(x). This conclusion confirms that the power series expansion of the function contains only even powers of x. The method employed for this proof is straightforward and effective, ensuring clarity in the demonstration of the function's properties.

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  • Understanding of even and odd functions
  • Familiarity with power series expansions
  • Knowledge of exponential functions, specifically e^x
  • Basic calculus concepts, including function evaluation
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  • Study the properties of even and odd functions in depth
  • Explore power series and their convergence criteria
  • Learn about the Taylor series expansion of functions
  • Investigate the applications of exponential functions in mathematical proofs
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Mathematicians, students studying calculus, and anyone interested in function properties and series expansions will benefit from this discussion.

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prove that x/(e^x-1) + x/2 is an even function, and therefore its power series only involves even powers of x.
 
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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.
 

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