Does anyone know how to code this summation equation that includes exponential and sine function?

[Dyuman Joshi]

See - syms and symsum (requires Symbolic Math Toolbox)

[Dyuman Joshi]

I do not know if the sum converges or not and I am feeling a bit lazy to check rn.

But symsum() or vpasum() don't seem to find an explicit value -

syms m t
PI = sym('pi');

term = 180/(2*m*PI)*exp(-(2*m)^2*PI^2*t/900)*sin(2*m*PI/30);

%symsum
out1 = symsum(term, m, 1, Inf)

out1 = 

vpa(subs(out1, t, 3))

ans = 

%vpasum
out2 = vpasum(term, m, 1, Inf)

out2 = 

vpa(subs(out2, t, 3))

ans = 

Answer 1

[Hassaan]

Calculates the summation of the series as described:

result = compute_summation(1,100); % Computes the series for x = 1 with 100 terms
disp(result);

   20.4225

function total = compute_summation(x, num_terms)
    if nargin < 2
        num_terms = 100; % Default number of terms
    end
    total = 0;
    for m = 1:num_terms
        exponential_part = exp(-(2*m)^2 * pi^2 / 900);
        sine_part = sin(2 * m * pi / 30);
        term = (180 / (2 * m * pi)) * exponential_part * sine_part;
        total = total + term;
    end
end

Note

Please note that computing an actual infinite series may not be possible due to computational limits, so you would typically compute a partial sum or find a closed-form if it exists.

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[Dyuman Joshi]

The series has an infinite summation, so the for loop will take too long for big values of num_terms (~1e9/1e10), nor will it give accurate results for bigger values of x.

And the exponential term is incorrect, as it is missing pi^2.

[Hassaan]

@Dyuman Joshi Thanks for pointing out I have corrected the exponential term. Also, I am not sure if MATLAB can do infinite series obviously not numerically nor symbolically dont we have to trancuate at some point?. Also I tried the syms approach.

syms m t;
% Define the general term of the series
general_term = (180/(2*m*pi)) * exp(-(2*m)^2 * pi^2 *t / 900) * sin(2*m*pi/30);

% Compute the summation
% Note: MATLAB might not be able to resolve this symbolically for an infinite series
% So we use a large number for the upper limit, say 100 terms
N = 100; % you can set N to a different value to approximate the infinite sum or inf
total_sum = symsum(general_term, m, 1, N);

% Display the symbolic sum
disp(total_sum);

% For numerical evaluation, substitute a value for x, for example x = 1
numerical_sum = subs(total_sum, m, 1);

% Display the numerical result
disp(vpa(numerical_sum));