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Calculus

Convergent and Divergent Sequences

         

If a sequence \[\begin{array} && a_n = \frac{ p n^2 + q n + 2}{3n + 2 } \end{array}\] converges to \( 2, \) what is the value of \( p +q? \)

A sequence \[ a_n = \frac{3 n^p + 3 n^{p-1} }{3 n^{q} - 2 n^{q-1}} \] converges to 0 as \( n\rightarrow\infty.\) Let the sequence \( b_n \) be \( \frac{1}{a_n}.\) Then which of the following is true about \(b_n\) as \(n\rightarrow\infty?\)

Which of the following sequences does not converge as \(n\rightarrow\infty:\) \[ \begin{array} &a_n=\sin{2 n}, &a_n=\sin{3 \pi n}, &a_n=\frac{\sin{5n}}{n} , &a_n=\cos{ 4 \pi n}?\end{array} \]

If a partial sum of the sequence \( a_n \) \[ \sum _{ n=k }^{ k+5 }{ a_n } \] converges as \( k\rightarrow\infty,\) what can we say about the sequence \( a_n? \)

If sequences \( a_n \) and \( b_n \) converge to \( 4 \) and \( 3 ,\) respectively, which of the following is true about the sequence: \[ (4 a_n -1 ) ( 6 b_n +2 ) ?\]

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