Chapter 8: Q15E (page 443)
Determine whether the series is convergent or divergent. If its convergent, find its sum.
\(\sum\limits_{n = 1}^\infty {\frac{{(n - 1)}}{{(3n - 1)}}} \)
The given series \(\sum\limits_{n = 1}^\infty {\frac{{(n - 1)}}{{(3n - 1)}}} \)isdivergent.
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Determine whether the sequence converges or diverges. If it converges, find the limit.
\({a_n} = \frac{{{{\left( { - 1} \right)}^n}n}}{{n + \sqrt n }}\)
Determine whether the series is convergent or divergent:\(\sum\limits_{n = 0}^\infty {\frac{{1 + \sin n}}{{{{10}^n}}}} \).
Use definition 2 directly to prove that \(\mathop {\lim }\limits_{n \to \infty } {r^n} = 0\)when\(\left| r \right| < 1\)
Find the first 40 terms of the sequence defined by\({a_{n + 1}} = \left\{ {\begin{aligned}{\frac{1}{2}{a_n}}&{{\rm{ if }}{a_n}{\rm{ is an even number }}}\\{3{a_n} + 1}&{{\rm{ if }}{a_n}{\rm{ is an odd number }}}\end{aligned}} \right.;{a_1} = 11\).
Do the same if\({a_1} = 25\). Make a conjecture about this type of sequence
Determine whether the sequence converges or diverges. If it converges, find the limit.
\({a_n} = \ln \left( {2{n^2} + 1} \right) - \ln \left( {{n^2} + 1} \right)\)
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