The following problem is from the American Mathematical Monthly. The problem only asks the reader to calculate it doesn’t give the answer; I added the answer myself.

**Problem **(Furdui, Romania). Let be real numbers with For an integer let

Let be the sign function. Show that

**Solution**. The characteristic polynomial of is

And roots of are

where

If is sufficiently large, which is the case we are interested in, then, since are either both positive or both negative (because ), and so So, in this case, are distinct real numbers and hence is diagonalizable in

Now is an eigenvector corresponding to if and only if if and only if Similarly, is an eigenvector corresponding to if and only if if and only if So if

then and hence

The rest of the solution is just Calculus and if you have trouble finding limits, see this post in my Calculus blog for details. We have