# Truncation Error Calculator

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

(2)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)We note that this inequality holds if the following inequality holds:

(4)### 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 $0.01$.