This is the proposal by teachers Marian Ursărescu and Florică Anastase, Romania and published by Romanian Mathematical Magazine.
Initially the answer of proposers agreed with me but later we had a slight different conclusions. As the per the proposers they arrived to .
If the answer of proposers is correct then probably I think I had done mistakes while evaluating the limits. I even tried on wolfram alpha it agrees with me ( doesn't provide the closed form but it approximates to my closed form) . Can anybody help me to correct my mistakes ?
Thank you ! :)
Here i wish to share my solution.
Firstly we recall Euler's formula and further solving gives where if is even and odd respectively. We note that latter sum is the imaginary part and hence equating we get
on setting and further expansion we get as the polynomial equation is of even degree and hence by Vieta's formula we have secondly we use the identity , taking on both side and on differentiating with respect to , we get that, ie . Replacing by we yield and we deduce that .
Call . As we can observe that we have limit form. Since as soon as ,function and with . So we can either make the direct use the formula for or without of it too. ie
. Here we have the limit of the form so we use L-hopital's rule to get
. And therefore,