Question

What is meant by the remainder $R_n$ of a series $\sum _{k=1}^{\infty }{a}_k$

Short answer

The remainder of the series $\sum _{k=1}^{\infty }{a}_k$ is $R_n=\sum _{k=n+1}^{\infty }{a}_k$.

Step-by-step solution

Step 1. Given Information.

The series:

$\sum _{k=1}^{\infty }{a}_k$

Step 2. The remainder of the convergent series.

If $\sum _{k=1}^{\infty }{a}_k$ is a convergent series with sum L, then we may approximate L with the nth partial sum $S_n=\sum _{k=1}^n{a}_k$. The nth remainder is defined by

$$\begin{gathered} R_n=L-S_n \\ =\sum _{k=n+1}^{\infty }{a}_k \end{gathered}$$