Calculus

Convergent and Divergent Sequences

Convergence - Problem Solving

         

If a sequence an=pn2+qn+23n+2\begin{array}{c}&& a_n = \frac{ p n^2 + q n + 2}{3n + 2 } \end{array} converges to 2, 2, what is the value of p+q? p +q?

A sequence an=3np+3np13nq2nq1 a_n = \frac{3 n^p + 3 n^{p-1} }{3 n^{q} - 2 n^{q-1}} converges to 0 as n. n\rightarrow\infty. Let the sequence bn b_n be 1an. \frac{1}{a_n}. Then which of the following is true about bnb_n as n?n\rightarrow\infty?

Which of the following sequences does not converge as n:n\rightarrow\infty: an=sin2n,an=sin3πn,an=sin5nn,an=cos4πn? \begin{array}{c}&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 an a_n n=kk+5an \sum _{ n=k }^{ k+5 }{ a_n } converges as k, k\rightarrow\infty, what can we say about the sequence an? a_n?

If sequences an a_n and bn b_n converge to 4 4 and 3, 3 , respectively, which of the following is true about the sequence: (4an1)(6bn+2)? (4 a_n -1 ) ( 6 b_n +2 ) ?

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