 Thus, \(\sum_{i=1}^\infty\frac{1}{2^i}=\lim_{n\to\infty}\sum_{i=1}^n\frac{1}{2^i}=\lim_{n\to\infty}(1-\frac{1}{2^n})=1\). We can show that if an infinite sequence of terms has a sum that converges, then the terms themselves must converge to 0. That’s because the terms \(\sum_{i=1}^n a_i\) must converge, so that for any \(\epsilon\), there must be \(N\), such that every one of the \(\sum_{i=1}^n a_i\) is within \(\epsilon\) of the limit whenever \(n&gt;N\). Thus, every such \(a_n\) must have absolute value less than \(2\epsilon\), so that both successive partial sums can be within \(\epsilon\) of this limit. (A similar condition holds also for infinite products, but from now on we will focus only on infinite sums.)