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The lecture rewrites the tail sum $R_n(s)$ by substituting $s=\sigma+it$, yielding terms of the form $\frac{1}{k^{\sigma}}e^{-it(\log k)}$. It then applies the triangle inequality, which states that the absolute value of a sum is at most the sum of the absolute values.

Conditions: $s = \sigma + it$ where $\sigma > 1$.; $R_n(s) = \sum_{k=n+1}^{\infty} \frac{1}{k^s}$ is the tail of the zeta series.; The triangle inequality is applied to the infinite series.

The lecture rewrites the tail sum $R_n(s)$ by substituting $s=\sigma+it$, yielding terms of the form $\frac{1}{k^{\sigma}}e^{-it(\log k)}$. It then applies the triangle inequality, which states that the absolute value of a sum is at most the sum of the absolute values.

Conditions: $s = \sigma + it$ where $\sigma > 1$.; $R_n(s) = \sum_{k=n+1}^{\infty} \frac{1}{k^s}$ is the tail of the zeta series.; The triangle inequality is applied to the infinite series.