Part 4/5:
This series captures all powers of ( x ) and is also alternating. The radius of convergence is ( R = 1 ), similar to the inverse tangent series.
Conclusion: Summary of Series and Convergence
To summarize the findings:
| Function | McLaren Series | Radius of Convergence |
|------------------|---------------------------------------------------|-----------------------|
| ( \frac{1}{1 - x} ) | ( \sum_{n=0}^{\infty} x^n ) | ( R = 1 ) |
| ( e^x ) | ( \sum_{n=0}^{\infty} \frac{x^n}{n!} ) | ( R = \infty ) |
| ( \sin x ) | ( \sum_{n=0}^{\infty} \frac{(-1)^n x^{2n+1}}{(2n+1)!} ) | ( R = \infty ) |