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## A root limit

Let be positive real numbers such that . Prove that

Solution

Without loss of generation , let . Then,

since forall . Thus, by the squeeze theorem it follows that

## A limit!

Evaluate the limit:

Solution

Let . Then,

It follows by Stolz–Cesàro that

In addition,

Hence .

## Floor series

Let denote the floor function. Evaluate the series

Solution

First of all we note that and are never squares. Thus, there exists a positive integer such that

It is easy to see that and thus we conclude that

Now is equal to the even number if-f

Hence, since the series is absolutely convergent we can rearrange the terms and by noting that the finite sums are telescopic , we get that:

## Double “identical” series

Compute the series

Solution

Successively we have:

## On the geometrical view of an integral

Evaluate the integral

using geometric methods.

Solution

We are working on the following figure

Thus,

since the red angle is due to the triangle since ( ). Therefore , the green angle is . Finally, the area of the circular sector is equal to

where and .

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