[math] \newcommand{\ex}[1]{\item } \newcommand{\sx}{\item} \newcommand{\x}{\sx} \newcommand{\sxlab}[1]{} \newcommand{\xlab}{\sxlab} \newcommand{\prov}[1] {\quad #1} \newcommand{\provx}[1] {\quad \mbox{#1}} \newcommand{\intext}[1]{\quad \mbox{#1} \quad} \newcommand{\R}{\mathrm{\bf R}} \newcommand{\Q}{\mathrm{\bf Q}} \newcommand{\Z}{\mathrm{\bf Z}} \newcommand{\C}{\mathrm{\bf C}} \newcommand{\dt}{\textbf} \newcommand{\goesto}{\rightarrow} \newcommand{\ddxof}[1]{\frac{d #1}{d x}} \newcommand{\ddx}{\frac{d}{dx}} \newcommand{\ddt}{\frac{d}{dt}} \newcommand{\dydx}{\ddxof y} \newcommand{\nxder}[3]{\frac{d^{#1}{#2}}{d{#3}^{#1}}} \newcommand{\deriv}[2]{\frac{d^{#1}{#2}}{dx^{#1}}} \newcommand{\dist}{\mathrm{distance}} \newcommand{\arccot}{\mathrm{arccot\:}} \newcommand{\arccsc}{\mathrm{arccsc\:}} \newcommand{\arcsec}{\mathrm{arcsec\:}} \newcommand{\arctanh}{\mathrm{arctanh\:}} \newcommand{\arcsinh}{\mathrm{arcsinh\:}} \newcommand{\arccosh}{\mathrm{arccosh\:}} \newcommand{\sech}{\mathrm{sech\:}} \newcommand{\csch}{\mathrm{csch\:}} \newcommand{\conj}[1]{\overline{#1}} \newcommand{\mathds}{\mathbb} [/math]

Determine whether or not each of the following infinite series converges, and evaluate it if it does.

• [a [math]\sum_{i=0}^\infty \frac7{5^i}[/math]
• [math]\sum_{k=1}^\infty \frac{a}{5^k}[/math]
• [math]\sum_{n=1}^\infty \left( \frac1{2^n} + \frac1n \right)[/math]
• [math]\sum_{j=0}^\infty \left(\frac1{2^j} - \frac1{3^j}\right)[/math]
• [math]\sum_{i=1}^\infty \frac{5\cdot2^i+6i}{i2^i}[/math]
• [math]\sum_{k=0}^\infty \left(3+\frac1{3^k}\right)[/math]
• [math]\sum_{i=1}^\infty \frac{i^2-1}{i^2+1}[/math]
• [math]\sum_{k=0}^\infty \frac{2^k+3^k}{6^k}[/math].