The question of whether or is larger is a classic one. Of course, with a calculator it is easy to see what the answer is. But how would you answer the question without a calculator?

There are lots of interesting ways to tackle the question. For this (short) post I would like to highlight one I saw on math.SE, given by Aryabhata. We need the assumptions that and for . (One way to view the latter claim is that, for , is larger than the first two terms of its Maclaurin series. Since all the rest of the terms of the Maclaurin series are positive, and the Maclaurin series converges to , we get that for .)

On to the proof. Since , . Applying the second assumption, then, we get . This means that , which in turn implies that .

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