Alternating Series Error Bound (College Board AP® Calculus BC): Revision Note

Alternating series error bound

What is an alternating series error bound?

• If an alternating series converges, the alternating series error bound places a bound on how far a partial sum (the sum of the first n terms) is from the value of the infinite series

• For an alternating series of the form $\sum _{n=1}^{\infty }{\left(-1\right)}^n·a_n$ or $\sum _{n=1}^{\infty }{\left(-1\right)}^{n+1}·a_n$

  • If the infinite series converges to a sum $S$

    • and if $s_n$ is the partial sum of the first n terms

  • Then $\left|S-s_n\right|\le a_{n+1}$

    • The absolute value of the difference between the sum of the infinite series, $S$, and the sum of the first n terms (the nth partial sum) $s_n$ is less than the absolute value of the next term in the series, $a_{n+1}$

      • Note that $a_{n+1}$ is the absolute value of the (n+1)th term

  • and the sign of $S-s_n$ is the same as the sign of that next term

    • If the next term is negative then $S-s_n\le 0\Rightarrow s_n\ge S$

    • or if the next term is positive then $S-s_n\ge 0\Rightarrow s_n\le S$

• As more and more terms are added to a converging alternating series, the sum of the terms will continually 'flip flop' above and below its infinite sum

  • However, it will keep getting closer and closer to that infinite sum

• An example is shown in the two tables below for the first 9 terms of the alternating harmonic series $\sum _{n=1}^{\infty }\frac{{\left(-1\right)}^{n+1}}{n}=\ln 2=0.693147...$

  • The $s_n$ column (sum of the first n terms) tends towards $\ln 2=0.693147...$

  • The $S-s_n$ column, the error between $\ln 2$ and $s_n$, tends towards zero

    • In the first table you can see how $s_n$ alternates between being above and below the value of the series sum

  • The absolute value of the numbers in the $S-s_n$ column are always less than the absolute value of the numbers in the (n+1)^th term column, i.e. $\left|S-s_n\right|\le a_{n+1}$