I know that Ramanujan Summation is used to assign a value to divergent series.
Such as $$\mathfrak R\biggl(\sum_{n=0}^\infty (-1)^n\biggr)=\frac{1}{2}$$
and $$\mathfrak R\biggl(\sum_{n=1}^\infty (-1)^nn\biggr)=-\frac{1}{4}$$
However, these values can also be obtained by using the geometric and power series, albeit outside of their radius of convergence.
$$\frac{1}{1+x}=\sum_{n=0}^\infty(-1)^nx^n,\;\;x=1$$
and $$-\frac{1}{(1+x)^2}=\sum_{n=1}^\infty (-1)^nnx^{n-1},\;\;x=1$$
Are there any cases where Ramanujan Summation does not coincide with the results obtained by plugging the value into a power series?
EDIT: Preferably where the power series is defined but does not match the Ramanujan Sum.
We have $$ \mathfrak R\left(\sum_{n=1}^\infty 1\right)= -\frac{1}{2}$$ but $$ \sum_{n=1}^\infty x^n = \frac{x}{1-x}$$ which diverges at $x=1.$
The method you are using is Abel summation, not Ramanujan.