Truncation Error Calculator

Posted on by
  1. How To Calculate Truncation Error
Truncation error formula for trapezoidal rule
Table of Contents

So far we have only looked at two methods for calculating the sums of convergent series. We have a simple technique for convergent Geometric Series, and we have a technique for Telescoping Series. We will now develop yet another technique which applies to convergent alternating series.

Theorem 1: Let $sum_{n=1}^{infty} a_n$ be a series satisfying all of the conditions of The Alternating Series Test, then we know that $sum_{n=1}^{infty} a_n = s$ for some $s in mathbb{R}$ (the series is convergent). The error estimation between the sum $s$ and the $n^{mathrm{th}}$ partial sum can be evaluated by using $mid s - s_n mid ≤ mid a_{n+1} mid = mid s_{n+1} - s_n mid$.
Truncation error calculator function

The theorem above tells us that if have a series that satisfies all of the conditions of the alternating series test, and we're given some allowed error, call it $E$, then we can determine the number of terms of the series $sum_{n=1}^{infty} a_n$ we must evaluated in order that our partial sum $s_n$ is within the error $E$ of the actual sum $s$. Let's look at some examples.

Truncation error calculator equation

To truncate a number, we miss off digits past a certain point in the number, filling-in zeros if necessary to make the truncated number approximately the same size as the original number. 01.03.2 Chapter 01.03 6. The number 1/10 is registered in a fixed 6 bit-register with all bits used for the fractional part. The difference gets accumulated every 1/10 th of a second for one day. The magnitude of the accumulated difference is Complete solution Page 29.

Example 1

Determine the number of terms of the series $sum_{n=1}^{infty} frac{(-1)^n}{n^2 + n}$ that are needed to be computed in order for the sum of the series to have an error less than $E = 0.001$.

Calculator

The series above satisfies all three conditions of the alternating series test (verify). Using the inequality above, we need to find an $n$ such that:

(1)
begin{align} quad mid s - s_n mid ≤ mid a_{n+1} mid = biggr rvert frac{(-1)^{n+1}}{(n+1)^2 + (n+1)} biggr rvert = frac{1}{n^2 + 3n + 2} < 0.001 end{align}

We note that this inequality holds only if the following inequality holds:

(2)
begin{equation} n^2 + 3n + 2 > 1000 end{equation}

We note that if $n = 31$, then $(31)^2 + 3(31) + 2 = 1060 > 1000$, and so if $n ≥ 31$ then $mid s - s_{n} mid < 0.001$, so the error between the partial sum $s_n$ and the actual sum $s$ is less than $0.001$.

Example 2

Determine the number of terms of the series $sum_{n=1}^{infty} frac{2(-1)^n}{n}$ that are needed to be computed in order for the sum of the series to have an error less than $E = 0.01$.

Once again this series satisfies all of the conditions of the alternating series test (verify), and so we need to find an $n$ such that the following inequality holds: Blackweb gaming keyboard software 3.1.

(3)
begin{align} quad mid s - s_n mid ≤ mid a_{n+1} mid = biggr rvert frac{2(-1)^{n+1}}{n+1} biggr rvert = frac{2}{n+1} < 0.01 end{align}
Truncation error formula

We note that this inequality holds if the following inequality holds:

(4)
begin{align} quad frac{n+1}{2} > 100 Leftrightarrow n > 199 end{align}

How To Calculate Truncation Error

So if $n ≥ 200$ then $mid s - s_n mid ≤ 0.01$ and so the error between the partial sum $s_n$ and the actual sum $s$ is less than $0.01$.