91Ó°ÊÓ

Give an example of: A power series that converges at \(x=5\) but nowhere else.

Give an example of: A finite geometric series with four distinct terms whose sum is 10

For the function \(f\) define a sequence recursively by \(x_{n}=f\left(x_{n-1}\right)\) for \(n>1\) and \(x_{1}=a .\) Depending on \(f\) and the starting value \(a,\) this sequence may converge to a limit \(L .\) If \(L\) exists, it has the property that \(f(L)=L .\) For the functions and starting values given, use a calculator to see if the sequence converges. [To obtain the terms of the sequence, repeatedly push the function button.] $$f(x)=\sin x, a=1$$

Give an example of: A series \(\sum C_{n} x^{n}\) with radius of convergence 1 and that converges at \(x=1\) and \(x=-1.\)

Explain what is wrong with the statement. The series \(\sum(1 / n)^{2}\) converges because the terms approach zero as \(n \rightarrow \infty.\)

Explain why the alternating series test cannot be used to decide if the series converges or diverges.$$\sum_{n=1}^{\infty}(-1)^{n-1} n$$

Give an example of: An infinite geometric series that converges to \(10 .\)

Explain why the alternating series test cannot be used to decide if the series converges or diverges.$$\sum_{n=1}^{\infty}(-1)^{n-1} \sin n$$

Which of the following geometric series converge? (I) \(20-10+5-2.5+\cdots\) (II) \(1-1.1+1.21-1.331+\cdots\) (III) \(1+1.1+1.21+1.331+\cdots\) (IV) \(1+y^{2}+y^{4}+y^{6}+\cdots,\) for \(-1

Let \(V_{n}\) be the number of new SUVs sold in the US in month \(n,\) where \(n=1\) is January \(2004 .\) In terms of SUVs, what do the following represent? (a) \(V_{10}\) (b) \(V_{n}-V_{n-1}\) (c) \(\sum_{i=1}^{12} V_{i}\) and \(\sum_{i=1}^{n} V_{i}\)