The ring of formal series in a variable $R=k[[x]]$ is a local ring, where the ideal maximal $m$ is the ideal of all series with term independent zero

Question

The ring of formal series in a variable $R=k[[x]]$ is a local ring, where the ideal maximal $m$ is the ideal of all series with term independent zero

I have already noticed this Maximal ideals of the ring of formal polynomials over a ring $R$ and I know that $k[[x]]$ is a local ring because $k$ field is a local ring, what I do not know is how to prove that the maximal ideal of $k[[x]]$ has that shape ($m$ is the ideal of all series with term independent zero.)

Answer 1

$(x)=xk[[x]]$ is a proper ideal consisting of elements with constant term $0$ (you agree, yes?)

$k[[x]]/(x)\cong k$ (you agree, yes?)

That last thing means $(x)$ is maximal. Since you already say you know the ring is local, then $(x)$ is the unique maximal ideal.