Chapter 8

The following recursively defined sequence can be used to compute \(\sqrt{k}\) for any positive number \(k .\) \(a_{1}=k ; a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right)\) This sequence was known to Sumerian mathematicians 4000 years ago, but it is still used today. Use this sequence to approximate the given square root by finding a \(6 .\) Compare your result with the actual value. $$\sqrt{21}$$

The following recursively defined sequence can be used to compute \(\sqrt{k}\) for any positive number \(k .\) \(a_{1}=k ; a_{n}=\frac{1}{2}\left(a_{n-1}+\frac{k}{a_{n-1}}\right)\) This sequence was known to Sumerian mathematicians 4000 years ago, but it is still used today. Use this sequence to approximate the given square root by finding a \(6 .\) Compare your result with the actual value. $$\sqrt{41}$$

Suppose that \(a_{n}\) and \(b_{n}\) represent arithmetic sequences. Show that their sum, \(c_{n}=a_{n}+b_{n}\) is also an arithmetic sequence.

Stacking Logs Logs are stacked in layers, with one fewer log in each layer. See the figure. If the top layer has 7 logs and the bottom layer has 15 logs, what is the total number of logs in the pile? Use a formula to find the sum.

Explain why the sequence \(\log 2, \log 4, \log 8, \log 16, \ldots\) is an arithmetic sequence.

Explain how we can distinguish between an arithmetic and a geometric sequence. Give examples.

Compare a sequence whose \(n\) th term is given by \(a_{n}=f(n)\) to a sequence that is defined recursively. Give examples. Which symbolic representation for defining a sequence is usually more convenient to use? Explain why.

The Natural Exponential Function The following series can be used to estimate the value of \(e^{a}\) for any real number a: $$e^{a} \approx 1+a+\frac{a^{2}}{2 !}+\frac{a^{3}}{3 !}+\dots+\frac{a^{n}}{n !}$$ where \(n !=1 \cdot 2 \cdot 3 \cdot 4 \cdot \cdots \cdot n\). Use the first eight terms of this series to approximate the given expression. Compare this estimate with the actual value. $$ e^{-1} $$

Use \(a_{k}\) and \(n\) to find \(S_{n}=\Sigma_{k-1}^{n} a_{k}\) (Refer to Example \(6 .\) ) Then evaluate the infinite geometric series \(S=\sum_{k=1}^{\infty} a_{k},\) Compare \(S\) to the values for \(S_{n^{}}.\) $$a_{k}=4\left(-\frac{1}{10}\right)^{k-1} ; n=1,2,3,4,5,6$$

Discuss the difference between a sequence and a series. Give examples.

Explain how to write the series $$ \log 1+\log 2+\log 3+\cdots+\log n $$ as one term.

Explain how to write the series \(\log 2-\log 4+\log 6-\log 8+\cdots+(-1)^{n-1} \log 2 n\) as one term. Assume \(n\) is even.