Mryntu

The following code

documentclassarticle
usepackage[utf8]inputenc
usepackage[english]babel

usepackageamsthm
usepackageamsmath
usepackage[left=1.5in, right=1.5in, top=0.5in]geometry


newtheoremdefinitionDefinition
newtheoremtheoremTheorem 
theoremstyleremark


begindocument
titleExtra Credit
maketitle

begindefinition
If f is analytic at $z_0$, then the series

$$f(z_0) + f'(z_0)(z-z_0) + fracf''(z_0)2!(z-z_0)^2 + cdots = sum_n=0^infty fracf^(n)(z_0)n!(z-z_0)^n hspace1cm(1)$$ 

is called the Taylor series for f around $z_0$.
enddefinition

begintheorem
If f is analytic inside and on the simple closed positively oriented contour $Gamma$ and if $z_0$ is any point inside $Gamma$, then
$$f^(n)(z_0) = fracn!2pi i int_Gamma fracf(zeta)(zeta - z_0)^n+1dzeta hspace1cm (n=1,2,3, cdots )$$ hfill (2)
endtheoremhrulefill

produces

How can I align the (1) and (2), and also get (2) to be on the same line as (n=1,2,3,...)?

I highly suggest to use a different approach:

documentclassarticle
usepackage[utf8]inputenc
usepackage[english]babel

usepackageamsthm
usepackageamsmath
usepackage[left=1.5in, right=1.5in, top=0.5in]geometry% I do not recommend to use this naiv canons of page construction for typographic reasons. 
usepackage[noabbrev]cleveref%new package


newtheoremdefinitionDefinition
newtheoremtheoremTheorem 
theoremstyleremark


begindocument
titleExtra Credit
maketitle

begindefinition
If f is analytic at $z_0$, then the series
beginalign%observe that empty line is removed
 f(z_0) + f'(z_0)(z-z_0) + fracf''(z_0)2!(z-z_0)^2 + cdots = sum_n=0^infty fracf^(n)(z_0)n!(z-z_0)^n labeleq:Taylor 
endalign%observe that empty line is removed
is called the emphTaylor series for f around $z_0$.%The definition is not in italics here to emphasize the term. 
enddefinition

begintheorem
If f is analytic inside and on the simple closed positively oriented contour $Gamma$ and if $z_0$ is any point inside $Gamma$, then
beginalign
 f^(n)(z_0) = fracn!2pi i int_Gamma fracf(zeta)(zeta - z_0)^n+1dzeta hspace1cm (n=1,2,3, cdots ).%every parenthesis should be ended with a dot. 
endalign
endtheorem

noindenthrulefill %alternative: hrule

You can use refer to the equation by: eqrefeq:Taylor or crefeq:Taylor. 

enddocument

Please read the comments in the code and What are the differences between $$, [, align, equation and displaymath?. Other useful staff is written in https://ctan.org/pkg/short-math-guide, https://ctan.org/pkg/lshort-english, and What are good learning resources for a LaTeX beginner?.

Use another approach, with the equation environment:

documentclassarticle
usepackage[utf8]inputenc
usepackage[english]babel

usepackageamsthm
usepackageamsmath
usepackage[left=1.5in, right=1.5in, top=0.5in]geometry


newtheoremdefinitionDefinition
newtheoremtheoremTheorem 
theoremstyleremark

begindocument
 titleExtra Credit
 maketitle

 begindefinition
 If f is analytic at $z_0$, then the series

 beginequation
 f(z_0) + f'(z_0)(z-z_0) + fracf''(z_0)2!(z-z_0)^2 + cdots = sum_n=0^infty fracf^(n)(z_0)n!(z-z_0)^n
 endequation

 is called the Taylor series for f around $z_0$.
 enddefinition

 begintheorem
 If f is analytic inside and on the simple closed positively oriented contour $Gamma$ and if $z_0$ is any point inside $Gamma$, then
 beginequation
 f^(n)(z_0) = fracn!2pi i int_Gamma fracf(zeta)(zeta - z_0)^n+1dzeta hspace1cm (n=1,2,3, cdots )
 endequation
 endtheorem
 hrulefill
enddocument