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Chapter 14: Problem 12
Write the first four terms of each sequence whose general term is given. $$a_{n}=\frac{(-1)^{n+1}}{2^{n}+1}$$
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Use a graphing utility to graph the function. Determine the horizontal asymptote for the graph of \(f\) and discuss its relationship to the sum of the given series. Function $$f(x)=\frac{4\left[1-(0.6)^{x}\right]}{1-0.6}$$ Series $$\begin{array}{l}4+4(0.6)+4(0.6)^{2} \\\\+4(0.6)^{3}+\cdots\end{array}$$
Factor: \(27 x^{3}-8\) (Section 6.4, Example 8)
Will help you prepare for the material covered in the next section. Consider the sequence whose \(n\) th term is \(a_{n}=4 n-3\) Find \(a_{2}-a_{1}, a_{3}-a_{2}, a_{4}-a_{3},\) and \(a_{5}-a_{4} .\) What do you observe?
Many graphing utilities have a sequence-graphing mode that plots the terms of a sequence as points on a rectangular coordinate system. Consult your manual; if your graphing utility has this capability, use it to graph each of the sequences in Exercises \(69-72 .\) What appears to be happening to the terms of each sequence as \(n\) gets larger? $$a_{n}=\frac{2 n^{2}+5 n-7}{n^{3}}, n:[0,10,1] \text { by } a_{n}:[0,2,0.2]$$
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The sum of the geometric series \(\frac{1}{2}+\frac{1}{4}+\frac{1}{8}+\dots+\frac{1}{512}\) can only be estimated without knowing precisely what terms occur between \(\frac{1}{8}\) and \(\frac{1}{512}\).
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