AS/A2 Pure 4 Pure 5

P6 Topic 4: Maclaurin and Taylor series
The power series for arctan(x)
The display shows the graph of y = arctan x. Also shown is the graph of the curve :
By using the slider you can vary the value of n from 1 to 20

This display shows the same information numerically. Keep x as 0.5, and hence find what value of n gives arctan 0.5 correct to 8 decimal places.

Summary Clearly, as n becomes bigger, the curve of the equation: becomes closer to the curve of y = arctan(x) - but only for values of x between -1 and 1. Convergence is quite fast. You will need to make n=4 and x=0.5 to get an approximation to arctan 0.5 that is correct to 3 decimal places. For 8 decimal place accuracy you will need to make n=10 and x=0.5. Based on free Java applets from JavaMath