Hfrhyu

The following code

documentclassarticle


usepackageamsthm
usepackageamsmath
usepackagemathtools

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



newtheoremdefinitionDefinition
newtheoremtheoremTheorem


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

begintheorem
(Cauchy's Integral Formula) Let $Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in some simply connected domain $D$ containing $Gamma$ and $z_0$ is any point inside $Gamma$, then
beginequation
f(z_0)= frac12pi i int_Gamma fracf(z)z-z_0 dz
endequation
endtheorem
noindent hrulefill

begintheorem
If f is analytic in the disk $|z-z_0|<R'$, then the Taylor series $(1)$ converges to $f(z)$ for all $z$ in this disk. 
endtheorem

produces the following image

How can I enclose Definition 1, Theorem 1, and Theorem 2 in separate rectangles. And have these rectangles separated by a space?

You can try with shadethm package, it can do all you want and many more. In you example what you need is:

documentclassarticle
usepackageshadethm
usepackagemathtools

newshadetheoremboxdefDefinition[section]
newshadetheoremboxtheorem[boxdef]Theorem
newtheoremtheorem[boxdef]Theorem

setlengthshadeboxsep2pt 
setlengthshadeboxrule.4pt 
setlengthshadedtextwidthtextwidth
addtolengthshadedtextwidth-2shadeboxsep
addtolengthshadedtextwidth-2shadeboxrule
setlengthshadeleftshift0pt 
setlengthshaderightshift0pt
definecolorshadethmcolorcmyk0,0,0,0
definecolorshaderulecolorcmyk0,0,0,1

begindocument

sectionBoxed theorems

beginboxdef
 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 $z_0$.
endboxdef

beginboxtheorem
 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
endboxtheorem

beginboxtheorem
(Cauchy's Integral Formula) Let $Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in some simply connected domain $D$ containing $Gamma$ and $z_0$ is any point inside $Gamma$, then
beginequation
f(z_0)= frac12pi i int_Gamma fracf(z)z-z_0 dz
endequation
endboxtheorem
noindent hrulefill

begintheorem
If f is analytic in the disk $|z-z_0|<R'$, then the Taylor series $(1)$ converges to $f(z)$ for all $z$ in this disk. 
endtheorem

enddocument

which produces the following:

Here is a solution with thmtools, which cooperates wit amsthm. Unrelated: you don't have to load amsmath if you load mathtools, as the latter does it for you:

documentclassarticle
usepackageamsthm, thmtools
usepackagemathtools

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

newtheoremdefinitionDefinition
newtheoremtheoremTheorem

declaretheorem[sibling=definition, shaded=rulecolor=black, rulewidth=0.6pt, bgcolor=rgb1,1,1,name=Definition]boxeddef
declaretheorem[sibling=theorem, shaded=rulecolor=black, rulewidth=0.6pt, bgcolor=rgb1,1,1,name=Theorem]boxedthm

begindocument
titleExtra Credit
author
maketitle

beginboxeddef
 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 $z_0$.
endboxeddef

beginboxedthm
 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
endboxedthm

beginboxedthm
(Cauchy's Integral Formula) Let $Gamma$ be a simple closed positively oriented contour. If $f$ is analytic in some simply connected domain $D$ containing $Gamma$ and $z_0$ is any point inside $Gamma$, then
beginequation
f(z_0)= frac12pi i int_Gamma fracf(z)z-z_0 dz
endequation
endboxedthm

noindent hrulefill

begintheorem
If f is analytic in the disk $|z-z_0|<R'$, then the Taylor series $(1)$ converges to $f(z)$ for all $z$ in this disk.
endtheorem

enddocument