Problem 57983. Easy Sequences 109: Summation of Derivatives of a Trigonometric Function

Ramon Villamangca

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A trigonometric function, ,  is defined as follows:
                
                where:     ; and 
                                x is in radians.
In this problem we are asked to evaluate the following summation:
                
                where:    x is a real number; 
                                n is an integer; 
                                 or the k-th derivative of T with respect to x; and 
                                .
For example for  and , we have:
  >>  syms x;
  >>  T = (1+2*tan(x)-tan(x)^2) / (1+tan(x)^2);
  >>  S = T/2^0 + diff(T)/2^1 + diff(T,2)/2^2;
  >>  s = vpa(subs(S,x,2))
        s = 0.10315887444431633673347091141408;
Please present the final output rounded-off to 6 decimal places. Therefore the final answer is .
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NOTE: Symbolic toolbox is not available to Cody players. It is possible to do numerical differentiation in Matlab without symbolic toolbox.  Just make sure that the output would be accurate within 6 decimal places of the 'exact' value obtained using syms, as shown above.