Computing e^(-2) Using a Power Series
Date: 04/16/2005 at 10:31:30 From: Anood Subject: Using power series to compute e^-2 Use power series to compute e^-2 to four decimal place accuracy. It's a very difficult question and I don't know how to do it.
Date: 04/18/2005 at 09:20:59 From: Doctor Ricky Subject: Re: Using power series to compute e^-2 Hey Anood, This problem isn't as bad as it looks, but you need to understand what it's asking. It's asking you for e^(-2), or in other words, if you are given: f(x) = e^(-x) you want the value of f(x) at x = 2. Therefore, you are going to find the MacLaurin series expansion for e^(-x). Do you remember the expansion for e^x, which is: x^2 x^3 x^n e^x = 1 + x + --- + --- + ... + --- + ... 2! 3! n! To find the expansion for e^(-x), just substitute -x in for x. Then we get: x^2 x^3 [(-1)^n]*(x^n) e^(-x) = 1 - x + --- - --- + ... + -------------- + ... 2! 3! n! Therefore to calculate e^(-2), we just plug the value x = 2 into our last equation. To determine how many terms we need to calculate e^(- 2) to four decimal places accurately, we calculate until the terms get small enough to where they are too small to change the value at four decimals. i.e. | [(-1)^n]*(x^n) | | -------------- | < 10^(-4) | n! | We use an absolute value since this is an alternating series. Therefore, we can rewrite it as: (x^n) 1 ----- < ------ n! 1000 Therefore, the number of terms (from zero to n) we will have to calculate is when: n! x^n < ------ 1000 We know x^n will be positive integers greater than one (since x = 2) and we can see that n! isn't bigger than 1000 until n = 7. Therefore we know we'll need at least 7 terms. By finding n! and x^n for larger values of n, we see that: n! x^n < ------ 1000 is true for the first time when n = 10. Therefore we need to calculate the first 11 terms of the MacLaurin series (remember that the first term is when n = 0). In other words, we would plug in our x = 2 into the MacLaurin series and use the first 11 terms (from x^0 until x^10) to get our answer accurate to four decimal places. If you have any questions or if this was a little confusing, please let me know! - Doctor Ricky, The Math Forum http://mathforum.org/dr.math/
Date: 04/19/2005 at 14:22:07 From: Anood Subject: Thank you (Using power series to compute e^-2) Dear Doctor Ricky, Thank you very very verrrrrrrrry much for your help! :>
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