Currently, I have a blank line appearing at the top of the page. However, I haven't inserted anything here to produce this line, nor is it a standardized format because other slides don't have this extra blank line either.
\begin{frame}[t]
\frametitle{Proof}
\begin{flalign*}
\displaystyle\sum_{k=0}^{n} b_k \alpha ^k &= 0 & \\
\displaystyle\overline{\sum_{k=0}^{n} b_k \alpha^k} &= \overline{0} \qquad
\text{Conjugating both sides} \\
\displaystyle\sum_{k=0}^{n} \overline{b_k \alpha^k} &=0 \qquad
\text{since }\overline{z+w}=\overline{z}+\overline{w} \\
\displaystyle\sum_{k=0}^{n} \overline{b_k} \; \overline{\alpha^k} &=0 \qquad
\text{since }\overline{zw}=\overline{z}\overline{w}\\
\displaystyle\sum_{k=0}^{n} \overline{b_k}\overline{\alpha}^k &= 0 \qquad
\text{since }\overline{z^k}=\overline{z}^k
\end{flalign*}
It comes out as this
Here is a minimal working document showing a slide with and without the blank space.
\documentclass[aspectratio=169]{beamer}
\mode<presentation> {
\usepackage{fancybox}
\usepackage{xcolor}
\usepackage{amsmath}
\usepackage{mdframed}
\usefonttheme{serif}
\usetheme{Madrid}
\usecolortheme{crane}
}
\usepackage{graphicx}
\usepackage{booktabs}
\begin{document}
\begin{frame}[t]
\frametitle{Proof}
\begin{mdframed}[backgroundcolor=lightgray]
\textbf{Given:}
The polynomial
$$P(x)=\sum_{k=0}^{n} b_k x^k$$ is a real polynomial with root $\alpha$ ie: $b_k \in \mathbb{R}$ and $P(\alpha)=0$.
\end{mdframed}
\begin{mdframed}[backgroundcolor=lightgray]
\textbf{Goal:}
Prove that $P(\bar{\alpha})=0$ ie: $$\sum_{k=0}^{n} b_k \bar{\alpha}^k=0.$$
\end{mdframed}
\end{frame}
\begin{frame}[t]
\frametitle{Proof}
\begin{flalign*}
\displaystyle\sum_{k=0}^{n} b_k \alpha ^k &= 0 & \\
\displaystyle\overline{\sum_{k=0}^{n} b_k \alpha^k} &= \overline{0} \qquad
\text{Conjugating both sides} \\
\displaystyle\sum_{k=0}^{n} \overline{b_k \alpha^k} &=0 \qquad
\text{since }\overline{z+w}=\overline{z}+\overline{w} \\
\displaystyle\sum_{k=0}^{n} \overline{b_k} \; \overline{\alpha^k} &=0 \qquad
\text{since }\overline{zw}=\overline{z}\overline{w}\\
\displaystyle\sum_{k=0}^{n} b_k\overline{\alpha}^k &= 0 \qquad
\text{since }\overline{z^k}=\overline{z}^k \text{ and the reals are preserved under complex conjugation }
\end{flalign*}
\end{frame}
\begin{frame}[t]
\frametitle{Proof}
\begin{align*}
\displaystyle\sum_{k=0}^{n} b_k \alpha^k &= 0 \\
\displaystyle\overline{\sum_{k=0}^{n} b_k \alpha^k} &= \overline{0}
\end{align*}
\end{frame}
\end{document}
Below is a 'good' slide.
