Suppose \(\sum_{n=1}^∞ a_n\) is a convergent series with positive terms. Let \(S_N\) be the \(N\)th partial sum of \(\sum_{n=1}^∞ a_n\), and let \(R_N\) be defined as:
\[ R_N = \sum_{n=N}^∞ a_n - S_N = \sum_{n=N+1}^∞ a_n\]
This means:
\[ \int^\infty_{N+1} f(x) dx \le R_N \le a_{N+1} + \int^\infty_{N+1} f(x) dx \]
Since \(\int^\infty_N f(x) dx \ge a_{N+1} + \int^\infty_{N+1} f(x) dx \):
\[ \int^\infty_{N+1} f(x) dx \le R_N \le \int^\infty_N f(x) dx \]
The remainder estimate.