# infsum(y,x) is infinite sum y;

infsum(a_+b_,x_):=infsum(a,x)+infsum(b,x);
#infsum(k_*y_,x_) := If(isfree(k,x), k*infsum(y,x));

#infsum((k_+x_)^a_, k_):=zeta(-a,x);

infsum(x_^(2k_+1)*1/(2k_+1)! *(-1)^k_,k_) := sin(x);
infsum(x_^(2k_)*1/(2k_)!*(-1)^k_,k_) := cos(x);
infsum((-1)^k_*e^(2i*k_*x_),k_):= -i/2*tan(x);
infsum(x_^(2k_+1)*1/(2k_+1)*(-1)^k_,k_) := atan(x);
infsum(x_^(2k_+1)/(2k_+1)!,k_) := sinh(x);
infsum(x_^(2k_)/(2k_)!,k_) := cosh(x);
infsum((-1)^k_*e^(2k_*x_),k_):= -1/2*tanh(x);
infsum(x_^(2k_+1)/(2k_+1),k_):=atanh(x);

infsum(x_^k_,k_):= when(abs(x)<1, 1/(1-x));
infsum(x_^(-k_),k_):= when(abs(x)>1, x/(x-1));
infsum(k_*x_^(k_-1),k_):= when(abs(x)<1,1/(1-x)^2);
infsum(x_^k_/k_,k_) :=  when(abs(x)<1, -log(1-x));
infsum((1-x_)^k_/k_,k_) := when(abs(x)<1, -log(x));
infsum((-1)^(k_+1)*x_^k_/k_,k_) := when(abs(x)<1, log(1+x));

infsum((a_+k_)^x_,k_):= zeta(a,-x);
#infsum(k_^x_,k_):= zeta(-x);
#infsum((-1)^k*k_^x_,k_):= eta(-x);
#infsum(x_^k_*k_^b_,k_):= polylog(-b,x);
#infsum(x_^k_*(a_+k_)^b_,k_):= polylog(a,-b,x);
#infsum((-1)^k_/k_,k_):= -log(2);

infsum(x_^k_/k_!,k_) := exp(x);
infsum(x_^(a_+k_)/k_!,k_) := x^a*e^x;
infsum(x_^(a_*k_)/k_!,k_) := exp(x^a);
infsum((-1)^k_/k_!*x_^k_,k_) := exp(-x);
infsum(k_/k_!*x_^k_,k_) := x*exp(x);
infsum(1/k_/k_!*x_^k_,k_) := ei(x)-log(x)-gamma;
infsum((-1)^k_*k_/k_!*x_^k_,k_) := x*exp(-x);
infsum(x_^k_/(a_*k_)!, k_):=mittag(a,x);
infsum(x_^k_/(a_*k_+b_)!, k_):=mittag(a,b-1,x);
infsum(x_^k_/gamma(a_*k_+b_), k_):=mittag(a,b,x);
infsum(1/k_!,k_):= e;
infsum(k_/k_!,k_):= e;
infsum(k_^2/k_!,k_):= 2e;
infsum(k_^3/k_!,k_):= 5e;
infsum((-1)^k_/k_!, k_) := exp(-1);

infsum(e^(a_*x_),x_) := when(a<0, exp(-a)/(exp(-a)-1));
infsum(exp(-k_*x_),k_):=1/(exp(x)-1);
infsum(e^(-x_),x_) := exp(1)/(exp(1)-1);
infsum(exp(-x_),x_) := exp(1)/(exp(1)-1);
infsum(x_*e^(-x_),x_) := exp(1)/(exp(1)-1)^2;

infsum(k_,k_):= infinity;

infsum(y_):= infsum(y,k);