Use x=-1/2 in the MacLaurin series for e^x to approximate 1/sqrt (e) to four decimal places.
So far I have
|f^n(0)/n! (1/sqrt(e)-1/2)^n)| </= 1/10^4, but I don't know what to do from here.
To approximate 1/sqrt(e) using the MacLaurin series for e^x, you can follow these steps:
1. Recall that the MacLaurin series for e^x is given by:
e^x = 1 + x + (x^2)/2! + (x^3)/3! + (x^4)/4! + ...
2. Substitute x = -1/2 into the series:
e^(-1/2) = 1 + (-1/2) + [(-1/2)^2]/2! + [(-1/2)^3]/3! + [(-1/2)^4]/4! + ...
3. Simplify each term:
e^(-1/2) = 1 - 1/2 + (1/2)^2/2! - (1/2)^3/3! + (1/2)^4/4! - ...
4. Continue simplifying each term by calculating the powers and factorials:
e^(-1/2) = 1 - 1/2 + 1/8 - 1/48 + 1/384 - ...
5. To approximate 1/sqrt(e), substitute x = -1/2 into the series and simplify further:
1/sqrt(e) ≈ 1 - 1/2 + 1/8 - 1/48 + 1/384 - ...
6. Continue adding terms until you reach the desired accuracy. In this case, you want to approximate 1/sqrt(e) to four decimal places, which means you need to find where the absolute value of the remaining terms is less than or equal to 1/10^4:
|remaining terms| = |(1/384)(1/2)^5/5! - (1/46080)(1/2)^6/6! + ...| ≤ 1/10^4
Keep adding terms until the absolute value of the remaining terms is less than or equal to 1/10^4.
7. Once you obtain the final approximation by adding the required terms, you will have your answer for 1/sqrt(e) to four decimal places.