Chapter 8: Q14RE (page 498)
Determine whether the series is convergent or divergent.
\(\sum\limits_{{\rm{n = 1}}}^\infty {{\rm{ln}}\left( {\frac{{\rm{n}}}{{{\rm{3n + 1}}}}} \right)} \)
The series is divergent.
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Show that the sequence defined by \({a_1} = 2,{a_{n + 1}} = \frac{1}{{3 - {a_n}}}\) sequences \(0 < {a_n} \le 2\) and is decreasing . Deduce that the sequence is covering and find its limit.
Calculate the first eight terms of the sequence of partial sums correct to four decimal places. Does it appear that the series is convergent or divergent?
\(\sum\limits_1^\infty {\frac{{{{( - 1)}^{n - 1}}}}{{n!}}} \)
Determine whether the series is convergent or divergent:\(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \).
Find the limits of the sequences\(\left( {\sqrt 2 ,\sqrt {2\sqrt 2 } ,\sqrt {2\sqrt {2\sqrt 2 } } ,........} \right)\).
Determine whether the sequence converges or diverges. If it converges, find the limit.
\({\bf{\{ 0,1,0,0,1,0,0,0,1, \ldots \ldots \ldots \ldots \ldots \} }}\)
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