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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

## 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 .