Mathematics

The remainder $R_n$ is the "tail" of the series—all the terms you haven't added yet. Since the terms are positive, decreasing, and continuous, you can trap $R_n$ between two integrals: the series tail sits above the integral from $n$ to $\infty$, and below the integral from $n-1$ to $\infty$. As $n$ grows, both bounding integrals shrink to zero, squeezing $R_n$ toward zero by the squeeze theorem.

The remainder $R_n$ is the "tail" of the series—all the terms you haven't added yet. Since the terms are positive, decreasing, and continuous, you can trap $R_n$ between two integrals: the series tail sits above the integral from $n$ to $\infty$, and below the integral from $n-1$ to $\infty$. As $n$ grows, both bounding integrals shrink to zero, squeezing $R_n$ toward zero by the squeeze theorem.