Matematické Fórum

Skrytý text:








Now, we present the method which can be used to derive such type of identities.

Step 1. Let us begin with a preliminary remark about the parameter . In virtue of the periodicity of the cosine-function we can consider only the case . Also, denote the given integral as follows:



It is easy to see that




Step 2. In the next we compute the indefinite integral (antiderivative):




Step 3. From the previous facts we can deduce that




Step 4. Thus



Now, we use the transformation



Moreover, one can deduce that




Hence,



This implies that




Step 5. Finally, in the last step we simplify the expression on the right hand side of the previous identity. Indeed, for every we have



Hence,



which implies the fact that



for every . It is possible to show that for the derived result holds also true. This finishes the proof of the desired identity.


Remark. In my computation there are several sub-steps which should be justified (the interchange of the order of integration, etc.). I hope, you can check this yourself.

Skrytý text:

1. It si obvious that the integral exists. First, we apply the Taylor expansion of :



a) Let and . Then and we can write that



It is not difficult to show that the series converges uniformely for given . Therefore it can be integrated, thus



Now we use the identity for the -st power of the cosine function (see):



If is even then



If is odd then



b) If and with then and



Now we use the similar technique with one difference. We find the integral



and then we determine the limit for . We derive from it that the blue idenitites hold true also in this case, i.e. for , .

2. We proved that



Let us find the sum of that series in a closed form.

3. Let us divide the series into two parts



Let us prove that both infinite series are convergent and find their sums. Then



It is not difficult to show that both series are uniformely convergent in .

4. Now, we will deal with the series . First, we derive it and then we use the binomial theorem for a real exponent (see)



5. Finally, we determine the sum of the series .



In this case, it is not so easy to find the value of as in the previous case. Let us define an auxilary function



Then . We will derive a special linear differential equation such that its solution is just the function .



So we get that



and that is a nonhomogeneous linear differential equation of the first order with the general solution


From the fact that we deduce that . Thus



We proved that