4

I have this equation:

enter image description here

I used the align in the middle part because I wanted the terms in the second sum to be aligned. But what I want is that the first sum, and the last double-sum, also start at the same place, how do I get that to work?

That is, I want all the sigma signs to start from the same place. But I also want to keep the alignment in the second sum that goes over multiple lines.

Can somebody please help? Here is the code:

\documentclass[a4paper,article]{memoir}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsmath}
\begin{document}
\begin{align*}
    &\sum\limits_{k=0}^\infty\left[(L\delta+k\delta)^{\underline{H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\right]^2\\ 
    =\sum\limits_{k=0}^\infty \Big[&(L\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &+((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &+((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-3)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &\vdots\\
    &+((L-(L-1))\delta+k\delta)^{\underline{ H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\Big]^2\\=&\sum\limits_{k=0}^\infty\left[\sum\limits_{j=1}^L ((j+k)\delta)^{\underline{H-\frac{1}{2}  }} -((j+k-1)\delta)^{\underline{H-\frac{1}{2}  }} \right]^2
    \end{align*}

\end{document}
6

here's a slightly different take on this display. i've used multline to shove the first line to the left, but since that doesn't really look balanced, i added some space at the beginning of the first line and the end of the aligned "substructure" to make it narrower.

\documentclass[a4paper,article]{memoir}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsmath}
\begin{document}
\begin{multline*}
    \hspace*{1cm}
    \sum_{k=0}^\infty\left[(L\delta+k\delta)^{\underline{H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\right]^2\\ 
\begin{aligned}
    =\sum_{k=0}^\infty \Big[&(L\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &+((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &+((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-3)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &\mathstrut\,\,\,\vdots\\
    &+((L-(L-1))\delta+k\delta)^{\underline{ H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\Big]^2\\
    =\sum_{k=0}^\infty\Bigg[&\sum_{j=1}^L ((j+k)\delta)^{\underline{H-\frac{1}{2}  }} -((j+k-1)\delta)^{\underline{H-\frac{1}{2}  }} \Bigg]^2
    \end{aligned}
    \hspace{1cm}
\end{multline*}

\end{document}

observe that, in a display, \limits isn't needed for sums, and the \left and \right on the last line have been changed to \Bigg to get around the &. i also added some space before the \vdots to center them under the plus signs.

enter image description here

3

There are probably better ways of doing this, but how does this look? I used an aligned for the second \sum.

enter image description here

\documentclass[a4paper,article]{memoir}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsmath}
\begin{document}
\begin{align*}
    &\sum\limits_{k=0}^\infty\left[(L\delta+k\delta)^{\underline{H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\right]^2\\ 
    ={}&\sum\limits_{k=0}^\infty \Big[
    \begin{aligned}[t]
    &(L\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &+((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &+((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-3)\delta+k\delta)^{\underline{H-\frac{1}{2} }}\\
    &\vdots\\
    &+((L-(L-1))\delta+k\delta)^{\underline{ H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\Big]^2
   \end{aligned}\\
={}&\sum\limits_{k=0}^\infty\left[\sum\limits_{j=1}^L ((j+k)\delta)^{\underline{H-\frac{1}{2}  }} -((j+k-1)\delta)^{\underline{H-\frac{1}{2}  }} \right]^2
    \end{align*}
\end{document}
3

This solution overlaps the alignment point using negative \hspace.

\documentclass[a4paper,article]{memoir}
\usepackage{amsthm}
\usepackage{amssymb}
\usepackage{amsmath}
\usepackage{mathtools}% for \vdotswithin
\usepackage{showframe}% MWE only
\begin{document}
\begin{align*}
    \sum\limits_{k=0}^\infty\left[(L\delta+k\delta)^{\underline{H-\frac{1}{2}  }}- (k\delta)^{ \underline{H-\frac{1}{2}   } }\right]^2
    \hspace{-3cm}& \\% overlap alignment point
    =&\sum\limits_{k=0}^\infty \Big[(L\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }} \\
    &\quad +((L-1)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }} \\
    &\quad +((L-2)\delta+k\delta)^{\underline{H-\frac{1}{2} }}-((L-3)\delta+k\delta)^{\underline{H-\frac{1}{2} }} \\
    &\quad \vdotswithin{+} \\
    &\quad +((L-(L-1))\delta+k\delta)^{\underline{ H-\frac{1}{2}  }}-(k\delta)^{ \underline{H-\frac{1}{2}   } }\Big]^2 \\
    =&\sum\limits_{k=0}^\infty\left[\sum\limits_{j=1}^L ((j+k)\delta)^{\underline{H-\frac{1}{2}  }} -((j+k-1)\delta)^{\underline{H-\frac{1}{2}  }} \right]^2
\end{align*}

\end{document}

demo

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