Part 3/5:
This series also exhibits an alternating pattern but includes even powers of ( x ). The radius of convergence remains ( R = \infty ).
Part E: The Function ( \tan^{-1} x )
The series for the inverse tangent function ( f(x) = \tan^{-1} x ) can be expressed as:
[
f(x) = \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{2n + 1}
]
Notably, this series mirrors the sine series structure but does not include factorials in the denominator. The radius of convergence for this function is ( R = 1 ), indicating convergence for ( |x| < 1 ).
Part F: The Function ( \ln(1 + x) )
The McLaren series for ( f(x) = \ln(1 + x) ) is given by:
[
f(x) = \sum_{n=1}^{\infty} \frac{(-1)^{n-1} x^n}{n}
]