There are given real number and real numbers and for all . Prove that:
.
Hint: Consider function near .
Thank you for your help in advance.
Ps: Happy Day!
There are given real number and real numbers and for all . Prove that:
.
Hint: Consider function near .
Thank you for your help in advance.
Ps: Happy Day!
The terms of the Taylor series of (for y>0) alternate in sign. It follows that . Therefore
Take exponentials to see that
So to prove the result, it suffices to show that
In one direction, this is fairly obvious: if the limit on the right is not zero then the greatest term in the sum on the left will not tend to 0, hence neither will the sum. For the other direction, notice that
Thus if , it follows that