Perhaps the representation of a trigonmetric function that has a value at pi, like cos x ?
The full question is: Determine the number of terms of the appropriate series needed to be added to approximate PI to five decimal places.
I believe I know the basics about power series, but I have no idea how to even start this. IF someone could help me with just that then I would be grateful.
yes but how do i begin to to do this; from my experience I believe that I must take the derivative until I see a pattern so that I can get a power series representation but then what. I'm pretty sure a remainder is involved but I don't know where or how??
A couple of points here
1. To generate the power series for arctan(x) you don't need to take derivatives
start with the series
for 1/(1-x)
by substitution of -x^2 for x you get the power series
for 1/(1+x^2)
Integrate this result to get series for arctan(x)
2. You now have an alternating series so use |S-Sn| < a(n+1)
to determine n.
I feel like this is all way above my head. I've seen everything thats been previously mentioned here and there but I juist can't put everything together. I have two more of these to do. Is there anyway someone could walk me through this problem so that I can solve the others. (I work best off of examples rather than hints that I can't combine with previous knowledge.)
Well note what was said above earlier, I may or may nor be correct, but I'll try my best to evaluate, above it was stated that , C26 mentioned the way to find the error for a alternating series for n iterations, now apply this to your problem, by finding the error of less than 5 decimals places, since you want to find pi within 5 decimals place, you will know by the the number iterations to take to approximate the error less than the tolerance you were given, assuming that your power series does go to some finite approximation of pi.
This is the way I see it, correct me If I am wrong
Ahh darn it, was replying while C26 was posting, sorry.
The series expansion of the 'arctan' function...
(1)
... in principle can be used to compute by the simple substitution in (1). However five decimals are required and that imposes the extimation of how many terms of (1) must be added to meet this goal. The series...
(2)
... is of the 'alternate sign' type and that means that the error in modulus is less than the last added term. So in order to obtain five exact decimal terms You have to sum about terms... not a properly little job! ...
If we remember that...
(3)
... we obtain...
(4)
Now setting in (1) we obtain...
(5)
... and the required precision can be achieved by summing terms... not bad! ...
Kind regards