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# Tag Archives: Real Analysis

## Arctan squared integral

Evaluate the integral

Solution

First of all we note that for

Hence by Parseval we get that

## Values of parameter

Find all values of such that

Solution

We’re invoking the same technique as in the problem hereSuccessively we have:

Obviously is differentiable in and its derivative is

It is . We distinguish cases:

• If then attains global maximum at equal to . Since it follows that .
• If then attains global minimum at equal to . In this case , however , the inequality cannot hold for all ; since is continuous its range is:

Hence this case is rejected.

• For the inequality obviously holds for all .

Summing up , .

## Convergence of a series

Examine the convergence of a series

Solution

From Taylor’s theorem with integral remainder we have that

However it is known that . Hence,

Exponentiating we get

Thus the series converges.

## Rational function and polynomial

Prove that there does not exist a rational function with real coefficients such that

where is a non constant polynomial.

Solution

Since polynomials are defined on we have that

Since tends to a finite value as it must be a constant polynomial. In particular, must be constant in the range of $$which is an infinite set, implying that must also be constant. This proves what we wanted.$$

## Factorial series

Evaluate the series

Solution

Let denote Kronecker’s delta and . We have successively,

It follows that