# Truncation Error Calculator

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1. How To Calculate Truncation Error 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\$. 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. 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\$. 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} 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\$.