AS/A2 Pure 4 Pure 5

P6 Topic 4: Maclaurin and Taylor series
Power series for the hyperbolic trig functions
Use the display to show the graph of y = sinh x, y = cosh x and y = arctanh x, together with power series approximations to these curves:
By using the slider you can vary the value of n from 0 to 20

This display shows the same information numerically for sinh x, cosh x and arctanh x. Note that sinh x is entered as (ex-e-x)/2 cosh x is entered as (ex+e-x)/2 arctanh x is entered as 0.5ln((1+x)/(1-x)) Keep x as 1, and hence find what value of n gives sinh 1 correct to 8 decimal places.

Summary Note that in the case of tanh x the power series approximation is only good for x between -1 and 1. For sinh x and cosh x, convergence is very fast. You will need to make n=2 and x=1 to get an approximation to sinh 1 that is correct to 3 decimal places. For 8 decimal place accuracy you will need to make n=5 and x=1. Based on free Java applets from JavaMath