- Whatis the truncationerrorof approximating by its fourth-degreeMaclaurinseriesin
for |x| ≤ 0.1.
CombiningSeries,EulersFormula
In manycases,we couldfind Taylor(Maclaurin)seriesof functionsfrom existingserieswithoutgoingthrough
the proofthat Examplesare products,quotientsand somesine and cosinefunctions.
Example 1 Find the Maclaurinseriesoff(x) =x sin xfor allx.
Solution.The Maclaurinseriesofsin xis. So the Maclaurinseriesofx sin xis
Example 2 Find the Maclaurinseriesoff(x) =cos^2 xfor allx.
Solution. We could avoid multiplying the Maclaurin series of cos x with itself, by applying:
on the Maclaurinseriesof , giving
For any real numberθ,eiθ=cos+i sin θwhere is the imaginaryunit. This is theEuler'sFormula.
Euler'sformulacombinesthe complementarysineandcosinefunctionsinto the simplerexponentialfunction
and heavilyappliesthe separationof real and imaginarypartsof complexnumbers.
Example 3 Find the Maclaurinseriesofcos xandsin xfor allxthrougheix.
Solution.cos x+i sin x=eixwhichhas a Maclaurinseries
by dividing into sum of odd and even indices. So
The Maclaurinseriesofcos xandsin xfollowby separatelytakingthe real and imaginaryparts.
Exercise1. Find and comparethe Maclaurinseriesforsin x cos xandsin 2x.