Calculus for Team-Based Inquiry Learning: 2022 Edition

(a)

Using Definition 8.3.12, given each restriction on $r\text{,}$ determine if ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}a{r}^n}$ converges.

1. $r>1\text{.}$

2. $r=1\text{.}$

3. $-1<r<1\text{.}$

4. $r=-1\text{.}$

5. $r<-1\text{.}$

(b)

Where possible, determine what value ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}a{r}^n}$ converges to.

(a)

What of the following approaches will best help us determine the convergence of this series?

1. This is a geometric series of the form ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}5\cdot {\left(\frac{1}{2}\right)}^n\text{.}}$

2. This is a geometric series of the form ${\displaystyle \sum _{n=0}^{\mathrm{\infty }}1\cdot {\left(\frac{1}{2}\right)}^n\text{.}}$

3. We can rewrite

   $$5+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots =4+(1+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots ).$$

(b)

Using your chosen approach, determine if $5+\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\cdots$ converges, and if so, to what it converges.

(a)

What of the following approaches will best help us determine the convergence of this series?

1. Noticing that ${\displaystyle 3\cdot \sum _{n=3}^{\mathrm{\infty }}2(0.7{)}^n=\sum _{n=0}^{\mathrm{\infty }}2(0.7{)}^n\text{.}}$

2. Noticing that ${\displaystyle 2+\sum _{n=3}^{\mathrm{\infty }}2(0.7{)}^n=\sum _{n=0}^{\mathrm{\infty }}2(0.7{)}^n\text{.}}$

3. Noticing that ${\displaystyle 2n+\sum _{n=3}^{\mathrm{\infty }}2(0.7{)}^n=\sum _{n=0}^{\mathrm{\infty }}2(0.7{)}^n\text{.}}$

4. Noticing that ${\displaystyle 2(0.7{)}^0+2(0.7{)}^1+2(0.7{)}^2+\sum _{n=3}^{\mathrm{\infty }}2(0.7{)}^n=\sum _{n=0}^{\mathrm{\infty }}2(0.7{)}^n\text{.}}$

5. Noticing that ${\displaystyle 0+1+2+\sum _{n=3}^{\mathrm{\infty }}2(0.7{)}^n=\sum _{n=0}^{\mathrm{\infty }}2(0.7{)}^n\text{.}}$

(b)

Using your chosen approach, determine if ${\displaystyle \sum _{n=3}^{\mathrm{\infty }}2(0.7{)}^n}$ converges, and if so, to what it converges.