Well, the LT is linear, correct? So you should be able to move the Laplace Transform operator inside the sum. What's the LT of the Dirac Delta function you have in there?
What is T? Is is positive? If so, you can lose the Heaviside step function (it'll just be 1 everywhere). What do you have on the LHS of this equation? Finally, assuming you can ignore the step function, you can rewrite the sum this way:
Does that suggest anything to you?
It's true that the sequence probably converges to zero. But that doesn't mean the series
converges to zero.