# Am I correct or is the book's answer correct

• freshman2013
In summary: I didn't bring it up because I don't know what a book would mean by "significant figures accurate". I would have thought that that would mean that the first four digits had to be correct. But that would make s6 the answer. This is why I think that the book is simply wrong.In summary, the sum of the series (-1)^(n-1)*(n^2)/(10^n) is 0.0676, accurate to 4 decimal places. This can be determined by using the Alternating Series Estimation Theorem or by evaluating the infinite summation exactly. The values of the partial sums, as computed by Maple, also support this answer. However, there may be a difference between 4-decimal

## Homework Statement

find the sum of (-1)^(n-1)*(n^2)/(10^n) correct to 4 decimal places

## The Attempt at a Solution

I got s5 since s6 is 36/1000000<0.0001
According to Wolframalpha, s5 is 0.06765 and s6 is 0.067614. Am i missing something here, or reading the problem wrong? The book claims it's s6 since s7 is 0.0000049 and that is the first value accurate to the 4th decimal place

freshman2013 said:

## Homework Statement

find the sum of (-1)^(n-1)*(n^2)/(10^n) correct to 4 decimal places

## The Attempt at a Solution

I got s5 since s6 is 36/1000000<0.0001
According to Wolframalpha, s5 is 0.06765 and s6 is 0.067614. Am i missing something here, or reading the problem wrong? The book claims it's s6 since s7 is 0.0000049 and that is the first value accurate to the 4th decimal place

To be accurate to 4 decimal places you want the first value in the sum that is smaller than 1/2 X 10-4. In other words, by adding this amount, it won't change the first four decimal places. 0.0001 is a bit too large, and could cause a change in the 4th decimal place.

freshman2013 said:
find the sum of (-1)^(n-1)*(n^2)/(10^n) correct to 4 decimal places
This is an alternating convergent series, so the error is bounded by the first omitted term. The sum is 0.06761833... To be accurate to four decimal places, you need to find where the first omitted term is less than this times 0.5*10-4, or 3.4*10-6. That's your s6. s5 is just a tad shy of delivering the required accuracy.

1.s5=.06765 and s6=.067614. The fourth digit after the decimal is 6, and it did not change between s5 and s6. Thus isn't s5 correct?

2.And why is it (1/2)*10^-4 instead of 1*10^-4? 0.0001 seems right to me since the biggest possible values smaller than that would be in the 0.00009... range and that's 4 zeros between the decimal point and a nonzero digit. More specifically, why the (1/2) in front? I would think it would be 1*10^-4 or 1*10^-5 or something like that. I think there's some concept that I'm completely missing or not understanding right now. Can someone explain this to me and give me examples and counterexamples?

freshman2013 said:
According to Wolframalpha, s5 is 0.06765 and s6 is 0.067614. Am i missing something here, or reading the problem wrong? The book claims it's s6 since s7 is 0.0000049 and that is the first value accurate to the 4th decimal place
You are mixing up the partial sums with the individual terms in the series.
S5 means the sum of the first 5 terms. What you call s7 would be better labeled as a7, the 7th term in the series. a6 is the first term that is smaller than .0001.

For your first question, I'll need to double check the values you got.
For the second question, if the first unused term is near .0001, adding it to the partial sum is likely to change the digit in the fourth decimal place. Adding a number that is smaller than .00005 (= .5 * 10-4) won't change what's in the fourth decimal place.

As for your first question, you are correct in saying that s5 gives the right answer to four decimal places. No doubt your textbook is using a theorem called something like Alternating Series Estimation Theorem, which is what D H cited. That theorem gives an upper bound on the error in truncating such a series. Since this is an upper bound, the actual error could be less.

1 person
freshman2013 said:

## Homework Statement

find the sum of (-1)^(n-1)*(n^2)/(10^n) correct to 4 decimal places

## The Attempt at a Solution

I got s5 since s6 is 36/1000000<0.0001
According to Wolframalpha, s5 is 0.06765 and s6 is 0.067614. Am i missing something here, or reading the problem wrong? The book claims it's s6 since s7 is 0.0000049 and that is the first value accurate to the 4th decimal place

You can use standard tricks to evaluate the infinite summation exactly; Maple gets
$$\sum_{n=1}^{\infty} (-1)^{n-1} \frac{n^2}{10^n} = \frac{90}{1331} \doteq 0.06761833208$$
Furthermore, you can also get the finite sums in closed form, but numerical methods are, of course, easier.

Here are s1..s10 as computed by Maple:
.1000000000, .6000000000e-1, .6900000000e-1, .6740000000e-1, .6765000000e-1, .6761400000e-1, .6761890000e-1, .6761826000e-1, .6761834100e-1, .6761833100e-1

Note that there is a difference between 4-decimal-place accuracy and 4 significant-figure accuracy; maybe the book means "significant figures" instead of "decimal places".

Ray Vickson said:
Note that there is a difference between 4-decimal-place accuracy and 4 significant-figure accuracy; maybe the book means "significant figures" instead of "decimal places".
That thought occurred to me as well.

## 1. Am I supposed to trust the book's answer or do my own research to verify it?

It is always beneficial to do your own research and verify information, especially in the field of science where new discoveries and advancements are constantly being made. However, textbooks and other educational resources typically go through a rigorous review process and are considered reliable sources of information.

## 2. How do I know if I am correct or if the book's answer is correct?

In science, there is often not just one correct answer. It is important to understand the reasoning behind a concept and be able to apply it in different scenarios. If your understanding aligns with the book's answer and can be demonstrated through experimentation or analysis, then you can consider yourself correct.

## 3. What should I do if I disagree with the book's answer?

If you disagree with the book's answer, it is important to analyze the evidence and arguments supporting both perspectives. You can also consult with your teacher or peers to discuss your reasoning and come to a consensus. Remember, in science, it is important to have an open mind and be willing to question and challenge information.

## 4. Is it okay to make mistakes when answering questions about science?

Yes, making mistakes is a natural part of the learning process in science. It is important to learn from your mistakes and use them as opportunities to improve your understanding. As long as you are actively working towards a better understanding and using reliable sources, making mistakes is a valuable learning experience.

## 5. How can I become more confident in my answers when studying science?

The best way to become more confident in your answers is to actively engage in the learning process. This includes reading, studying, asking questions, and practicing problems. It is also helpful to review and summarize key concepts and check your understanding through self-assessments. As you become more familiar with the material, your confidence will naturally increase.