Verify Maclaurin's infinite series expansion of $$\frac{1}{1+x}=1-x+x^2-x^3+\dots$$ for $-1< x< 1$.
The problem is to check the convergence of the series in $-1< x< 0$ and $0\leq x\leq 1$ by finding the remainder term $$R_n=\frac{x^n}{n!}f^n(\theta x) ~(\text{Lagrange's form})$$ and $$R_n=\frac{x^n(1-\theta)^{n-1}}{(n-1)!}f^n(\theta x) ~(\text{Cauchy's form})$$, $0<\theta<1$.
I have the following result also. If $y=\frac{1}{1+x}$,then $y_n=D^ny=(-1)^nn!(x+1)^{-n-1}$.
I have two questions:
1. How to check $\lim_{n\to \infty} R_n=0$ for $-1< x< 0$ and $0\leq x< 1$. (Note: PROBLEM: For Cauchy type remainder: $R_n$ has one $n$ in the numerator. How to take limit to show $\lim_{n\to \infty} R_n=0$)
2. How to decide when Lagrange's form is required to be calculated and when Cauchy's form.