Mrthdrb

This is what I have typed so far in latex:

documentclass[10pt,letter]article


usepackageamsmath
usepackageamsthm
usepackagemathtools


usepackagegraphicx


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


onehalfspacing



begindocument



titleHomework Chapter 5

author




maketitle 



section*Section 5.2: 4,5,7

paragraph4.
Let $alpha$ be a complex number. Show that if $(1+z)^alpha$ is taken as $e^alpha operatornameLog(1+z)$, then for $|z|< 1$\

$(1+z)^alpha = 1 + displaystylefracalpha1z + fracalpha(alpha -1)1cdot2z^2 + fracalpha(alpha-1)(alpha-2)1cdot2cdot3z^3 + cdots$\

In general, 

beginequation

displaystylefracd^jdz^j(1+z)^alpha=fracalpha !(1+z)^alpha - j(alpha - j)!

endequation

I get an error with the following code:

beginequation

displaystylefracd^jdz^j(1+z)^alpha=fracalpha !(1+z)^alpha - j(alpha - j)!

endequation

I was wondering if someone could clarify why this happens?

The error is caused by the blank lines inside the equation environment: no blank line should appear there.

There are also several “non fatal” errors in your code.

1. Never end paragraphs in text with \, but with a blank line.




2. Use displaystyle very sparingly: I believe to have used it in the body of a document just a handful of times. For the special occasion where you need a “big fraction” in text mode, there is dfrac. In the first case you use it, it is rather equation* than an inline equation.




3. Using operatornameLog is fine, but I guess that using Log after defining DeclareMathOperatorLogLog is better, because in a complex analysis context the symbol will be quite often used.




4. As a subtlety, remember to add , after ! if something immediately follows a factorial.




5. cdots and ldots are fine, but just dots is in general better. However, when dots is not between alike symbols, you need to help amsmath; in your case, use dotsb.

I also reorganized a bit your preamble.

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

usepackageamsmath
usepackagemathtools
usepackageamsthm

usepackagegraphicx
usepackagesetspace

onehalfspacing

DeclareMathOperatorLogLog

begindocument

titleHomework Chapter 5
author

maketitle 

section*Section 5.2: 4,5,7

paragraph4.
Let $alpha$ be a complex number. Show that if $(1+z)^alpha$ is taken 
as $e^alpha Log(1+z)$, then for $|z|< 1$
beginequation*
(1+z)^alpha = 1 + fracalpha1z + 
fracalpha(alpha -1)1cdot2z^2 +
fracalpha(alpha-1)(alpha-2)1cdot2cdot3z^3 + dotsb
endequation*
In general, 
beginequation
fracd^jdz^j(1+z)^alpha=
fracalpha! , (1+z)^alpha - j(alpha - j)!
endequation

enddocument

Do you see that the input is simpler? You don't have to worry where line ends: it's TeX that generally takes care of that.

In addition to getting into the habit of not creating blank lines inside display-math environments (such as equation), do also try to get used to not treating LaTeX like it's some kludgy version of MS Word, i.e., like it's a word processor. What do I mean by that? Basically, get over your instincts of applying lots and lots of visual formatting. Instead, get used to writing what you mean -- and let LaTeX handle the nitty-gritty formatting issues.

It takes a couple of minutes (ok, a couple of hours...) to get used to not treating LaTeX like it's a word processor. But once you get the hang of it, you'll really start to enjoy LaTeX much more. Here's how I would re-write your code to make it more "LaTeX-y":

Note, for instance, the use of [ and ] to start and end an unnumbered display-math environment. Learning not to use \ to force line breaks (other than in tabular and array environments) is particularly important. And don't leave blank lines (which tell TeX to start a new paragraph) immediately before [ or beginequation. The amsmath package also provides the instruction DeclareMathOperator, which is handy for creating new "operators" such as Log.

documentclass[letterpaper]article % not 'letter' 
usepackageamsmath
DeclareMathOperatorLogLog
usepackageenumitem % allows fine-tuning of 'enumerate' environment
begindocument

titleHomework Chapter 5
author
date
maketitle

setcountersection5 % just for this example
setcountersubsection1

subsectionProblems 4, 5, and 7

beginenumerate[wide=0pt, label=bfseriesarabic*., start=4]

item Let $alpha$ be a complex number. Show that if $(1+z)^alpha$ is
taken as $e^alphaLog(1+z)$, then for $|z|< 1$,
[
(1+z)^alpha = 1 + fracalpha1 z + fracalpha(alpha-1)1cdot2 z^2
 + fracalpha(alpha-1)(alpha-2)1cdot2cdot3 z^3 + dotsb
]
In general,
beginequation labeleq:taylor_exp
fracd^jdz^j (1+z)^alpha
 =fracalpha! (1+z)^alpha - j(alpha - j)!
endequation
dots

item dots

stepcounterenumi

item dots

endenumerate

enddocument