This is a very classic exercise and can be dealt with various ways. We know the result in advance. Why? Because it is the Taylor Polynomial of the exponential function. Let us see however how we gonna deal with it with High School Methods.
Find the positive real number such that
Define the function
and note that forall . Clearly , is differentiable and its derivative is given by
It follows that . Suppose that . Then the monotony of as well as the sign of is seen at the following table.
It follows then that . This is an obscurity due to the fact that . Similarly, if we suppose that . Hence
For we easily see that the given inequality holds.