Problems in Geometric Series
Compute $5+\frac{10}{3}+\frac{20}{9}+\frac{40}{27}+ \ldots$
Compute $5-\frac{10}{3}+\frac{20}{9}-\frac{40}{27}+ \ldots$
Compute $\sum\limits_{n=2}^{9} 2^{3n} 3^{-2n}$
Compute $\sum\limits_{n=2}^{\infty} 2^{3n} 3^{-2n}$
Compute $\sum\limits_{n=-\infty}^{-2} 2^{-3n} 3^{2n}$. See if you can substitute $m = -n$ and obtain the expression in the previous problem
Simply $\sum\limits_{n=2}^{\infty} x^n 3^{-n}$ an expression as a rational function of $x$. Evaluate this function for $x=2$
Compute $\sum\limits_{n=1}^{\infty} \cos^n (\pi t)$ and express as a function of $t$
Compute $\sum\limits_{n=1}^{\infty} \frac{1}{2^n} \cos(n \pi t)$ and express as a function of $t$. Hint: Use Euler's formula to convert this sum of two geometric series.
Simplify $\sum\limits_{n=1}^{\infty} \left(\frac{1}{3} \right)^n e^{j \omega n}$
Compute $e^{j\frac{\pi}{2}}+\frac{1}{2} e^{{j \pi}} + \frac{1}{4} e^{j\frac{3 \pi}{2}} + \ldots + \frac{1}{2^9} e^{j\frac{10 \pi}{2}}$. Simplify your answer into a complex number in Cartesian form
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