Chapter 8: Q19E (page 463)
Determine whether the series is absolutely convergent, conditionally convergent,or divergent.
\(\sum\limits_{n = 1}^\infty {\frac{n}{{{5^n}}}} \)
The Series is Absolutely Convergent..
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Determine whether the sequence converges or diverges. If it converges, find the limit.
\({{\rm{a}}_{\rm{n}}} = \frac{{\sin 2n}}{{1 + \sqrt n }}\)
(a) Let,\({a_1} = a\),\({a_2} = f\left( a \right)\),\({a_3} = f\left( {{a_2}} \right) = f\left( {f\left( a \right)} \right)\), . . . ,\({a_{n + 1}} = f\left( {{a_n}} \right)\), where\(f\)is a continuous function. If\(\mathop {\lim }\limits_{n \to \infty } {a_n} = L\), show that\(f\left( L \right) = L\).
(b) Illustrate part (a) by taking\(f\left( x \right) = \cos x\),\(a = 1\), andestimating the value of\(L\)to five decimal places.
\(\sum\limits_{n = 2}^\infty {\frac{3}{{n\left( {n + 3} \right)}}} \)
Prove that if \(\mathop {\lim }\limits_{n \to \infty } {a_n} = 0\)and {\({b_n}\)} is bounded, then \(\mathop {\lim }\limits_{n \to \infty } ({a_n}{b_n}) = 0.\)
Determine whether the series is convergent or divergent. If its convergent, find its sum.
\(\sum\limits_{k = 1}^\infty {\frac{{k(k + 2)}}{{{{(k + 3)}^2}}}} \)
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