Trjtdtk

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. %This is the usual approach to refer to formulas. 

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

One of the key features of (La)TeX is its ability to automatically number sections, theorems, list items and so on for you. This includes equations! Using

beginequation
labelsomename
 e=mc^2
endequation

creates an equation with the next number in sequence, and you can refer back to it with eqrefsomename. Just as with other automatically numbered thing, you can insert a new numbered equation before this one and all the numbers will be correctly updated. You can create unnumbered equations with the equation* environment (or, with less typing, [ ... ]). Other equation-like environments also have a * version that doesn't number.

Note that it is not recommended to use $$...$$ for equations in LaTeX.

(Also, I'd strongly recommend numbering all Definitions, Theorems, etc. in the same series. It's really annoying to be looking for Theorem 4 in a long document when seeing Lemma 3 and Definition 5 give you no hint about whether you should look backwards or forwards.)