 However, just having the terms \(a_n\) converge to 0 is not sufficient for the sum to converge. A deep and important fact about infinite sums is that \(\sum_{i=1}^\infty\frac{1}{i}\) fails to converge, because the partial sums eventually exceed any finite bound. To see this, note that the first term is greater than \(1/2\), the next two terms are both greater than \(1/4\), the next four terms are all greater than \(1/8\), and in general there are \(2^{n-1}\) terms that are each greater than \(1/2^n\). So to get a partial sum greater than \(n\), it is sufficient to add the first \(2^{2n}\) terms.