The first series does not converge. As the sequence continues, the numbers will continue to get larger and larger, going above 1 whole. Adding numbers like this together does not approach a numeric limit, but instead approaches infinity. The second series does converge. As the sequence continues, the numerator will stay 1 and the denominator will continue to increase. This means the overall fraction gets smaller and smaller, approaching 0. Thus the sum will approach a finite number. The third series does not converge. As the exponents get larger and larger, the answers will swing to larger positives and larger negatives each time. As the sum goes on, the absolute value of it will get larger and the sign will switch. The fourth series does not converge. As 2 gets raised to a higher and higher exponent, the product of it and 1/5 will be a larger and larger number. It will approach infinity, not a finite number.