For $α ∈ ℝ$ the function $g_α \colon B_1(0) → ℝ, x ↦ (1+x)^α$ is $C^∞$ and $g_α^{(n)}(x) = n! \tbinom{α}{n}(1+x)^{α-n}$, where $\tbinom{α}{n} = \frac{α(α-1)\cdots(α-n+1)}{n!}$ is the generalized binomial coefficient.
I want to show that $g_α$ has a Taylor expansion $g_α (x) = \sum_{k=0}^∞ \tbinom{α}{k}x^k$. I can easily show that the series really converges for $|x| < 1$, but can I use this fact to show that the remainder term
$$R_n(x) = \int_0^x \frac{(x-t)^n}{n!} (n+1)! \tbinom{α}{n+1}(1+t)^{α-n-1} dt$$
or in Lagrange form
$$\quad R_n(x) = \frac{x^{n+1}}{(n+1)!} (n+1)! \binom{α}{n+1}(1+ξ)^{α-n-1} = x^{n+1} \binom{α}{n+1}(1+ξ)^{α-n-1}$$
converges to zero for $|x| < 1$?
How else can I show that the remainder term converges to zero for $|x| < 1$?
Progress: Using the Lagrange form, for $x ∈ (-1,1)$ and $n ∈ ℕ$ there is an $h=h(n,x) ∈ (0,1)$ such that:
$$|R_n(x)| = \Big| x^{n+1} \binom{α}{n+1}(1+hx)^{α-n-1} \Big| = \Big| \binom{α}{n+1}\big(\frac{x}{1+hx}\big)^{n+1}(1+ξ)^{α} \Big| $$
If $x > 0$, then $\big|\frac{x}{1+hx}\big| < {\big|\frac{x}{1+x}\big|}< 1$ and so for $q := \big|\frac{x}{1+x}\big|$, since $\sum_{k=0}^∞ \tbinom{α}{n} q^n$ converges:
$$|R_n(x)| ≤ \Big| \binom{α}{n+1} · q^{n+1} ·(1+ξ)^{α} \Big| \overset{n → ∞}{\longrightarrow} 0$$
The problem is still open for $x < 0$.
Mhenni suggested to use Stirling approximation, but I want to do this more elementary and I think it is possible.
Best Answer
Again, I think that I got it. One needs to proceed differently:
Let $x ∈ ℝ$ such that $|x| < 1$. For $t ∈ ℝ$ between $x$ and zero, one has $0 ≤ |t| ≤ |x| < 1$ as well as $|x-t| < |1+t|$, because:
Therefore $\big|\frac{x-t}{1+t}\big|$ assumes a maximum $q<1$ for $t$ on the compact interval between zero and $x$. Then: \begin{align*} |R_n(x)| &= \Big| \int_0^x \frac{(x-t)^n}{n!} g_α^{(n+1)}(t) dt \Big|\\ &≤ \Bigg| \int_0^x \Big| \frac{(x-t)^n}{n!} g_α^{(n+1)}(t) \Big| dt \Bigg| \\ &\overset{(1)}{=} \Bigg| \int_0^x \Big| \frac{(x-t)^n}{(1+t)^n} ·(α-n) · \binom{α}{n} · (1+t)^{α-1} \Big| dt \Bigg|\\ &\overset{(2)}{≤} n · \binom{α}{n} ·|q|^n · \Bigg| \int_0^x |α/n -1| ·|1+t|^{α-1} dt\Bigg| \\ &\overset{(3)}{≤} n · \binom{α}{n} · |q|^n · (|α| +1) · C \overset{n → ∞}{\longrightarrow} 0, \end{align*} where the convergence follows from $|(n+1)\tbinom{α}{n+1}\big/n\tbinom{α}{n}|\overset{n → ∞}{\longrightarrow} 1$ and $|q| < 1$, and