How Many Terms Needed for Accurate Series Estimation with Less Than 0.001 Error?

In summary, the conversation is about a question on a test regarding finding the number of terms needed to estimate a sum with an error of less than .001 using the alternating series remainder theorem. One person believes the answer is 48 terms while the other believes it is 47 terms. The expert summarizes that the correct answer is 48 terms, but it is important to approach the teacher humbly in order to resolve the issue.
  • #1
CraftyVisage
2
0

Homework Statement



Okay, well this was a question on one of my recent tests:

How many terms do you have to use to estimate the sum from n = 0 to n = infinity of
(-e/pi)^n with an error of less than .001?

Homework Equations



Alternating series remainder theorem:

For an alternating series.

The absolute value of (S - S(n) = (Rn) = (error) is less than or equal to a(n+1)

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

The Attempt at a Solution



My solution: Using the alternating series remainder theorem, says that absolute value of (S-Sn)= (error) is less than or equal to a(n+1)

I find that a(48) is the first term less than .001 it is proximately equal to .000961. This means that my series must include all terms up to a(47) to have an error that is guaranteed to be less than .001. Since the series starts at 0 this means my total sum = a(0)+a(1)+a(2)+...+a(45)+a(46)+a(47). This gives me a total of 48 terms and the answer to the question is 48.

My math teacher claims that the answer is 47 total terms. I think that the answer cannot possibly be 47 terms. This means that your highest term is a(46). So again according to the alternating series remainder theorem:

error < or = a(n+1) or a(47). a(47) is a proximately equal to .00111

.00111 is not less than .001 so you are not guaranteed that your error is actually less than .001, it only has to be less than .00111.

Who is correct? Is the answer 48. Or am I just missing something and the answer is 47?
 
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  • #2
ttt please help. I kept bugging my teacher about this and he got so angry that he threatened to kick me out of class permanently. My parents believe I am crazy and having someone who knows their stuff chime in on this would put it to rest.
 
  • #3
CraftyVisage said:
ttt please help. I kept bugging my teacher about this and he got so angry that he threatened to kick me out of class permanently. My parents believe I am crazy and having someone who knows their stuff chime in on this would put it to rest.

I would say you are correct; there are 48 terms required. The value of n required is [itex]n=47[/itex], but since n starts from 0, this produces 48 terms.

The best way to talk to your teacher about this is to (1) apologize to your teacher and (2) present your proof (it should be at least as well-written as what you have hear) and ask your teacher if he can help you better understand the problem by "pointing out your mistake." Even though I do not believe you are in err, by being humble and making yourself out to be the one at fault (even when you are not), you will find your teacher appreciative of the opportunity to save face.
 

Related to How Many Terms Needed for Accurate Series Estimation with Less Than 0.001 Error?

What is an alternating series?

An alternating series is a series of numbers in which the terms alternate in sign between positive and negative. For example, 1 - 2 + 3 - 4 + 5 - ... is an alternating series.

What is the Alternating Series Test?

The Alternating Series Test is a test used to determine the convergence or divergence of an alternating series. It states that if the terms in an alternating series decrease in absolute value and approach 0, then the series is convergent.

How do you find the remainder of an alternating series?

To find the remainder of an alternating series, you can use the Alternating Series Remainder formula:|Rn| ≤ |an+1|, where Rn is the remainder, an+1 is the next term in the series, and n is the number of terms in the series. This formula gives an upper bound for the remainder of the series, meaning the actual remainder will be less than or equal to this value.

Can an alternating series diverge?

Yes, an alternating series can diverge. The Alternating Series Test only tells us that if the terms decrease in absolute value and approach 0, then the series is convergent. If these conditions are not met, the series may diverge.

How is the remainder of an alternating series related to the convergence of the series?

The remainder of an alternating series is related to the convergence of the series because it gives an approximation of the error in the series. If the remainder is small, it means that the series is converging towards a specific value. If the remainder is large, it may indicate that the series is diverging or converging towards a different value.

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