We develop an efficient simulation algorithm for computing the tail probabilities of the infinite series $S = \sum_{n \geq 1} a_n X_n$ when random variables $X_n$ are heavy-tailed. As $S$ is the sum of infinitely many random variables, any simulation algorithm that stops after simulating only fixed, finitely many random variables is likely to introduce a bias. We overcome this challenge by rewriting the tail probability of interest as a sum of a random number of telescoping terms, and subsequently developing conditional Monte Carlo based low variance simulation estimators for each telescoping term. The resulting algorithm is proved to result in estimators that a) have no bias, and b) require only a fixed, finite number of replications irrespective of how rare the tail probability of interest is. Thus, by combining a traditional variance reduction technique such as conditional Monte Carlo with more recent use of auxiliary randomization to remove bias in a multi-level type representation, we develop an efficient and unbiased simulation algorithm for tail probabilities of $S$. These have many applications including in analysis of financial time-series and stochastic recurrence equations arising in models in actuarial risk and population biology.