Section 1
Sigma Notation
Evaluate.$$\sum_{k=0}^{2}(3 k+1)$$.
Evaluate.$$\sum_{k=1}^{4}(3 k-1)$$.
Evaluate.$$\sum_{k=0}^{3} 2^{k}$$.
Evaluate.$$\sum_{k=1}^{4} \frac{1}{2^{k}}$$.
Evaluate.$$\sum_{k=0}^{3}(-1)^{k} 2^{k}$$.
Evaluate.$$\sum_{k=0}^{3}(-1)^{k} 2^{k+1}$$.
Evaluate.$$\sum_{k=2}^{4} \frac{1}{3^{k-1}}$$.
Evaluate.$$\sum_{k=3}^{5} \frac{(-1)^{k}}{k !}$$.
Evaluate.$$\sum_{k=0}^{3}\left(\frac{1}{2}\right)^{2 k}$$.
Evaluate.$$\sum_{k=0}^{3}(-1)^{k}\left(\frac{1}{2}\right)^{2 k}$$.
Express in sigma notation.$$1+3+5+7+\dots+21$$.
Express in sigma notation.$$1-3+5-7+\cdots-19$$.
Express in sigma notation.$$1 \cdot 2+2 \cdot 3+3 \cdot 4+\dots+35 \cdot 36$$.
Express in sigma notation.$$\text { The lower sum } m_{1} \Delta x_{1}+m_{2} \Delta x_{2}+\cdots+m_{n} \Delta x_{n}$$.
Express in sigma notation.$$\text { The upper sum } M_{1} \Delta x_{1}+M_{2} \Delta x_{2}+\cdots+M_{n} \Delta x_{n}$$.
Express in sigma notation.$$\text { The Riemann sum } f\left(x_{1}^{*}\right) \Delta x_{1}+f\left(x_{2}^{*}\right) \Delta x_{2}+\cdots+f\left(x_{n}^{*}\right) \Delta x_{n}$$.
Write the given sums as $\sum_{k=3}^{10} a_{k}$ and as $\sum_{i=0}^{7} a_{i+3}.$$$\frac{1}{2^{3}}+\frac{1}{2^{4}}+\dots+\frac{1}{2^{10}}$$.
Write the given sums as $\sum_{k=3}^{10} a_{k}$ and as $\sum_{i=0}^{7} a_{i+3}.$$$\frac{3^{3}}{3 !}+\frac{4^{4}}{4 !}+\dots+\frac{10^{10}}{10 !}$$.
Write the given sums as $\sum_{k=3}^{10} a_{k}$ and as $\sum_{i=0}^{7} a_{i+3}.$$$\frac{3}{4}-\frac{4}{5}+\dots-\frac{10}{11}$$.
Write the given sums as $\sum_{k=3}^{10} a_{k}$ and as $\sum_{i=0}^{7} a_{i+3}.$$$\frac{1}{3}+\frac{1}{5}+\frac{1}{7}+\dots+\frac{1}{17}$$.
Transform the first expression into the second by a change of indices.$$\sum_{k=2}^{10} \frac{k}{k^{2}+1} ; \quad \sum_{n=-1}^{7} \frac{n+3}{n^{2}+6 n+10}$$.
Transform the first expression into the second by a change of indices.$$\sum_{n=2}^{12} \frac{(-1)^{n}}{n-1} ; \quad \sum_{k=1}^{11} \frac{(-1)^{k+1}}{k}$$.
Transform the first expression into the second by a change of indices.$$\sum_{k=4}^{25} \frac{1}{k^{2}-9} ; \quad \sum_{n=7}^{28} \frac{1}{n^{2}-6 n}$$.
Transform the first expression into the second by a change of indices.$$\begin{aligned} &\sum_{k=0}^{15} \frac{3^{2 k}}{k !}\\ &81 \sum_{n=-2}^{13} \frac{3^{2 n}}{(n+2) !} \end{aligned}$$
Express the decimal fraction $0 . a_{1} a_{2} \cdots a_{n}$ in sigma notation using powers of $1 / 10.$
Show that $\sum_{k=1}^{n} \frac{1}{\sqrt{k}} \geq \sqrt{n}.$
Use a graphing utility or CAS to evaluate the sum.$$\sum_{k=0}^{50} \frac{1}{4^{k}}$$.
Use a graphing utility or CAS to evaluate the sum.$$\sum_{k=1}^{50} \frac{1}{k^{2}}$$.
Use a graphing utility or CAS to evaluate the sum.$$\sum_{k=0}^{50} \frac{1}{k !}$$.
Use a graphing utility or CAS to evaluate the sum.$$\sum_{k=0}^{50}\left(\frac{2}{3}\right)^{k}$$.