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! (Nod)

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- March 14th 2011, 09:54 AMzadirProving equivalency (limits)
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! (Nod) - March 15th 2011, 01:01 PMOpalg
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