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## Binomial coefficients as multiple sum

Prove that

Solution

The binomial coefficient in the RHS enumerates the subsets of size of . The LHS does the same thing, but choosing first the largest element of , then its second-to-largest element , until choosing its smallest element .

## Trigonometric integral

Prove that

Solution

Let be the integral. Note that

and hence:

For the integral we apply the substitution . Then, and

(1)

and similarly by applying the change of variables at the second integral we get that

(2)

Adding equations , we get that

and the result follows.

## Parametric integral

Let . Prove that:

Solution

We’re applying the change of variables and thus,

## Abel’s Integral

Prove that:

Solution

We are applying Abel – Plana. We choose thus,

## A convergent series

Let be a convergent series of positive terms. Prove that there exists a strictly increasing sequence which is also unbounded such that the series also converges.

Solution

We set and we observe that

That’s all folks.

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