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Problems #5,6 are graded.

Problem 5: Let $\{a_k\}_{k=0}^{\infty}=\left\{ \dfrac{1}{3^k} \right\}_{k=0}^{\infty}$ and $\{b_k\}_{k=0}^{\infty} = \left\{ \dfrac{1}{7^k} \right\}_{k=0}^{\infty}$. Then compute using geometric series: $$\begin{array}{ll} \left\langle \{a_k\}, \{b_k\} \right\rangle &= \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{3^k} \dfrac{1}{7^k} \\ &= \displaystyle\sum_{k=0}^{\infty} \left( \dfrac{1}{21} \right)^k \\ &= \dfrac{1}{1-\frac{1}{21}} \\ &=\dfrac{21}{20}. \end{array}$$ Now let $\{c_k\}_{k=0}^{\infty} = \left\{ \sqrt{\dfrac{1}{k!}} \right\}$. Using the fact that $e^x = \displaystyle\sum_{k=0}^{\infty} \dfrac{x^k}{k!}$ we see that $e^1=e=\displaystyle\sum_{k=0}^{\infty} \dfrac{1}{k!}.$ Now we may compute $$\begin{array}{ll} \left\langle \left\{ \sqrt{\dfrac{1}{k!}} \right\}, \left\{ \sqrt{\dfrac{1}{k!}} \right\} \right\rangle &= \displaystyle\sum_{k=0}^{\infty} \sqrt{\dfrac{1}{k!}}\sqrt{\dfrac{1}{k!}} \\ &= \displaystyle\sum_{k=0}^{\infty} \dfrac{1}{k!} \\ &= e. \end{array}$$ Problem 6: Let $\vec{x}=5+4i$ and $\vec{y}=21+16i$. Compute $$\begin{array}{ll} \left\langle \vec{x},\vec{y} \right\rangle &= (5+4i)\overline{(9-11i)} \\ &=(5+4i)(9+11i) \\ &=45 + 55i + 36i + 44i^2 \\ &= 45 + 91i - 44 \\ &= 1 + 91i. \end{array}$$ Let $z_1=21+16i$ and notice $$\begin{array}{ll} z_2&=\dfrac{11-5i}{2+i} \\ &=\dfrac{11-5i}{2+i} \dfrac{2-i}{2-i} \\ &= \dfrac{22 - 11i -10i + 5i^2}{4 - i^2} \\ &= \dfrac{17-21i}{5} \\ &= \dfrac{17}{5} - \dfrac{21}{5}i. \end{array}$$ Now compute $$\begin{array}{ll} \left\langle z_1,z_2 \right\rangle &= (21+16i) \overline{ \left( \dfrac{17}{5} - \dfrac{21}{5}i \right)} \\ &= (21+16i) \left( \dfrac{17}{5} + \dfrac{21}{5} i \right) \\ &= \dfrac{357}{5} + \dfrac{441}{5}i + \dfrac{272}{5}i + \dfrac{336}{5}i^2 \\ &=\dfrac{21}{5} + \dfrac{713}{5}i. \end{array}$$