# Series expansion of a function up to linear terms [duplicate]

I have the following:

\[CapitalSigma] = r^2 + a^2 Cos[\[Theta]]^2;
\[CapitalDelta] = r^2 - 2 M r + a^2 - k/3 r^2 (r^2 + a^2);
grr = \[CapitalSigma]/\[CapitalDelta];


and I want to obtain the expansion of grr up to linear terms in $$M/r$$ and $$k r^2$$. What I have done is to employ Series but I could not get the right answer which is $$1+\frac{2M}{r}-\frac{kr^2}{3}$$ Does anyone have an idea as to what I am missing here? Thanks

• Does this answer your question? Multivariable Taylor expansion does not work as expected Commented Aug 16, 2021 at 16:30
• Why is there no dependence on a? Commented Aug 16, 2021 at 16:34
• That also bothers me @CarlWoll, although the instruction is to expand up to linear terms in $M/r$ and $kr^2$ only Commented Aug 16, 2021 at 16:41
• @CarlWoll, I am only expanding with respect to $r$ for large $r$. Commented Aug 16, 2021 at 16:48
• Is the required true? Limit[grr, r -> Infinity] produces 0. Commented Aug 16, 2021 at 16:50

Assuming k r^2, M/r, a/r to be small of same order, expansion gives nearly the result you're expecting:
grr /. {M -> r eps M/r , k -> 1/r^2 eps k r^2,a -> r  eps a /r } // Simplify

• Pay your attention to the minus sign - k r^2/3 in the required expression. Commented Aug 16, 2021 at 18:50
• Ok , I think you need to define the order of parameter a too. Otherwice it isn't possible to do the series expansion. Commented Aug 17, 2021 at 6:26