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

[CraftyVisage]

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?

 

[CraftyVisage]

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.

 

[rs1n]

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 $n=47$, 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.