E[min (t, sigma y(i))] >= Sigma [q(i)/p(i)]

or in other words

E[min (t,i1+i2+i3+.........+in)*Pr(i1)Pr(i2)...........P r(in)]

i1,i2,i3,,,,,,,in goes from 1 to infinity

where y is a random variable following geometric distribution.

Is there a way to find a lower bound for t which would make the computation much easier.

Even for the case of i=2 i.e. y1+y2 the computation gets huge. Let alone for solving for a general n here. We have the input of