Proofs for Taylors theorem and other forms Let $f \in C^k[a,b]$.Show that for $x,x_0 \in [a,b]$,
$$f(x)=\sum\limits_{j=0}^\mathbb{k-1}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+{1\over k!}{\int_{x_0}^x f^{(k)}(t)(x-t)^k \,dt}$$
and after this use this result for proving the following forms of Taylor's theorem:
$i)$ if we assume the strongest condition of derivative $f \in C^{k+1}[a,b]$ show that
$$f(x)=\sum\limits_{j=0}^\mathbb{k}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+O(x-x_0)^{k+1}$$
$ii)$ if we do not have condition of higher derivative show that
$$f(x)=\sum\limits_{j=0}^\mathbb{k}{{1\over j!}f^{(j)}(x_0)(x-x_0)^j}+o(x-x_0)^{k+1}$$
I am trying to find a complete answer(proof) for this parts of Taylors theorem but i did not manage to find anything since on internet.Any proof for this would be apreciated.

Set 
$$
\varphi(x)=\sum_{j=0}^{k-1}\frac{f^{(j)}(x)}{j!}(z-x)^j
$$
then
$$
φ'(x)=\frac{f^{(k)}(x)}{(k-1)!}(z-x)^{k-1}
$$
and thus
$$
φ(z)-φ(x)=\int_x^z\frac{f^{(k)}(t)}{(k-1)!}(z-t)^{k-1}\,dt
$$
Now change the variables $x$ to $x_0$ and then $z$ to $x$ to obtain
$$
f(x)=\sum_{j=0}^{k-1}\frac{f^{(j)}(x_0)}{j!}(x-x_0)^j+\frac{1}{(k-1)!}\int_{x_0}^x f^{(k)}(t)(x-t)^{k-1}\,dt
$$
So it seems the powers and factorials in the error term in your first formula are somewhat off.

For the conclusions, take the (second) most simple case as illustration. 
$$
f(x)=f(x_0)+f'(\tilde x)(x-x_0),\qquad \tilde x\in [x_0,x],
$$
gives directly $f(x)=f(x_0)+O((x-x_0)^1)$ and with a virtual zero
\begin{align}
f(x)&=f(x_0)+f'(x_0)(x-x_0)+\Bigl(f'(\tilde x)-f(x_0)\Bigr)(x-x_0)\\
&=f(x_0)+f'(x_0)(x-x_0)+o((x-x_0)^1)
\end{align}
for $f\in C^1(\Bbb R)$.
