syms r t m rp
q = int(rp*sin(m*pi*r),rp,0,1); % the integral r prime from 0 to 1
V = symsum(exp(m*pi*t)*sin(m*pi*r)*q, m, 1, 10) % a summation
Finally, the V(r,t) obtained:
(exp(pi*t)*sin(pi*r)^2)/2 + (exp(2*pi*t)*sin(2*pi*r)^2)/2 + (exp(3*pi*t)*sin(3*pi*r)^2)/2 + (exp(4*pi*t)*sin(4*pi*r)^2)/2 + (exp(5*pi*t)*sin(5*pi*r)^2)/2 + (exp(6*pi*t)*sin(6*pi*r)^2)/2 + (exp(7*pi*t)*sin(7*pi*r)^2)/2 + (exp(8*pi*t)*sin(8*pi*r)^2)/2 + (exp(9*pi*t)*sin(9*pi*r)^2)/2 + (exp(10*pi*t)*sin(10*pi*r)^2)/2
