# Approximating Alternating Series w/ 0.0000001 Accuracy: 4 Terms Needed

In summary, the question asks for the number of terms needed in order for the partial sum to be within 0.0000001 of the convergent value of the given series. The answer is 4 terms, as the fourth term is the first to be smaller than the permitted remainder. This can be determined by calculating the values of each term and noting their progression in size.
$$\sum_{n=1}^{\infty} a_n = 1 - \frac {(0.3)^2}{2!} + \frac {(0.3)^4}{4!} - \frac {(0.3)^6}{6!} + \frac {(0.3)^8}{8!} - ...$$

how many terms do you have to go for your approximation (your partial sum) to be within 0.0000001 from the convergent value of that series?

the answer to this question is 4, but i don't know how the book got 4. Probably a real easy question, but I am really confuse since there are no examples i can find, so can someone help? i really don't even know where to start, but i found this:

$$|s-s_n| \leq |s_n+1 - s_n| = b_n +1$$

any help will be appreciated

$$\sum_{n=1}^{\infty} a_n = 1 - \frac {(0.3)^2}{2!} + \frac {(0.3)^4}{4!} - \frac {(0.3)^6}{6!} + \frac {(0.3)^8}{8!} - ...$$

how many terms do you have to go for your approximation (your partial sum) to be within 0.0000001 from the convergent value of that series?

the answer to this question is 4, but i don't know how the book got 4. Probably a real easy question, but I am really confuse since there are no examples i can find, so can someone help? i really don't even know where to start, but i found this:

$$|s-s_n| \leq |s_n+1 - s_n| = b_n +1$$

any help will be appreciated

Calculate the values of each of the terms and note the progression of the sizes of them. Isn't that inequality supposed to be

$$|s-s_n| \leq |s_{n+1} - s_n| = b_{n +1}$$

It is saying that the absolute value of the remainder after n terms will be no greater than the absolute value of the difference between the sum to n + 1 terms and the sum to n terms. Another way of saying that is look at the next term.

http://www.mathwords.com/a/alternating_series_remainder.htm

For an alternating series, you only have to look at the magnitude of the first term you are dropping from the sum to estimate the remainder.

OlderDan said:
Calculate the values of each of the terms and note the progression of the sizes of them.

well the 4th term is $$\frac{(0.3)^6}/{6!}$$ but it comes out to .000001

and the 5th term has 8 zeros, so the 4th term is closer to the value 0.0000001. so is that how the book got 4th term as an answer?

The question is asking to find $n$ such that:

$$s - s_n < 10^{-7}$$

Once youve found the first n, there's no need to go further.

well the 4th term is $$\frac{(0.3)^6}/{6!}$$ but it comes out to .000001

and the 5th term has 8 zeros, so the 4th term is closer to the value 0.0000001. so is that how the book got 4th term as an answer?

It is not a question of being closer. It is a question of greater than or lesser than. The fourth term is ten times bigger than the permitted remainder, so you have to keep it. You would have to keep it even if its value were .00000010000. . .000001. The first term you can leave out is the first term that is smaller than .00000001. That is the fifth term. That is how the book got the answer.

## What is an alternating series?

An alternating series is a series where the terms alternate in sign. For example, the series 1 - 1/2 + 1/3 - 1/4 + 1/5 -... is an alternating series.

## Why is it important to approximate alternating series with high accuracy?

Approximating alternating series with high accuracy is important because these series often do not have a finite sum and thus must be approximated to a certain level of precision. This allows for a better understanding of the behavior of the series and its convergence or divergence.

## What does "4 terms needed" mean in the context of approximating alternating series with 0.0000001 accuracy?

In this context, "4 terms needed" means that in order to approximate the series with an accuracy of 0.0000001, the first 4 terms of the series must be calculated and summed.

## How is the accuracy of an alternating series approximation determined?

The accuracy of an alternating series approximation is determined by calculating the difference between the actual sum of the series and the approximated sum. This difference should be within the desired accuracy, in this case, 0.0000001.

## What is the mathematical formula for approximating an alternating series with 0.0000001 accuracy using 4 terms?

The formula for approximating an alternating series with 0.0000001 accuracy using 4 terms is: s~4~ = a~1~ + a~2~ + a~3~ + a~4~, where s~4~ is the approximate sum, and a~1~, a~2~, a~3~, and a~4~ are the first 4 terms of the series.

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