The given series looks like the series which converges, so we'll guess that the given series converges too. In order to convince the teacher, we have to find a series Σ *b*_{n} that converges and has bigger terms than the given series. Since making the denominator smaller makes the whole fraction bigger, it follows that Then since the series converges and has bigger terms than the original series, the comparison test says that the original series must converge also. Since limits of summation don't affect whether a series converges, it's okay if the relationship 0 < *a*_{n} ≤ *b*_{n} doesn't hold for all the terms at the beginning, so long as it holds for all the terms from some point onwards. |