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\title{Power Series}
\date{November 2, 2020}

\begin{document}

\ 

\vspace{-.25in}

\maketitle

\vspace{-.15in}

\thm{If $\lvert u \rvert < 1$, then
%\[1 + x + x^2 + x^3 + x^4 + \dots = 
$\displaystyle \sum_{n=0}^{\infty}  u^n = \frac{1}{1-u} $.
}


\vspace{.1in}


\prob{[Completion]
Find the sum of the series (a function of $x$) and its radius of convergence: \[ 1 - 3x + 9x^2 - 27x^3 + 81x^4 - \dots\]
}


\vspace{.1in}

\prob{[Graded]
Find the sum of the series (a function of $x$) and its radius of convergence: \[ 3x + 6x^3 + 12x^5 + 24x^7 + 48x^9 + \dots \]
}

\vspace{.1in}


\prob{[Completion]
Find the sum of the series (a function of $x$) and its radius of convergence: \[ \sum_{n=0}^{\infty} \left[ 2(x-5)\right]^n\]
}

\vspace{.1in}


\prob{[Completion]
Find the radius of convergence and interval of convergence of the power series: $\displaystyle \sum_{n=1}^{\infty} \frac{(x+2)^n}{n}$
}

\vspace{.1in}


\prob{[Graded]
Use the root test to find the radius of convergence of the power series: $\displaystyle \sum_{n=0}^{\infty} \left(\frac{x - 7}{2}\right)^{2n}$ 
}


\vspace{.1in}


\prob{[Completion]
Use the ratio test to find the radius of convergence of the power series: $\displaystyle \sum_{n=1}^{\infty} \frac{(x-2)^n}{n!}$ 
}

\end{document}