There are a number of methods of determining whether a series converges or diverges. More generally, the sequence formula 27 converges linearly with rate formula 6 if formula 29. It is the sequence of probability distributions that converges. Finally, the sequence formula 32 converges sublinearly and logarithmically. This estimator converges in probability to formula 37. This shows that the martingale converges with probability formula 45. It converges with probability 1 to that underlying distribution. An improper integral converges if the limit defining it exists. Hence, by definition, the series formula 12 converges uniformly. By definition, in a Hilbert space any Cauchy sequence converges to a limit. The trapezoidal rule converges rapidly for periodic functions. In that case, the computation eventually converges to the correct output. Johannes Kepler observed that the ratio of consecutive Fibonacci numbers converges. In particular, if is an integer greater than 1, then this series converges. When formula 66, the probability of win converges to formula 67 ().