Chapter 8: Q10RE (page 498)
Determine whether the series is convergent or divergent \(\sum\limits_{{\rm{n = 1}}}^\infty {\frac{{{{\rm{n}}^{\rm{2}}}{\rm{ + 1}}}}{{{{\rm{n}}^{\rm{3}}}{\rm{ + 1}}}}} \).
The series is perfectly Diverges because the result is positive\({\rm{1}}\).
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(a)Show that if \(\mathop {\lim }\limits_{n \to \infty } {a_2}_n = L\)and \(\mathop {\lim }\limits_{n \to \infty } {a_{2n + 1}} = L,\) then {\({a_n}\)} is convergent and \(\mathop {\lim }\limits_{n \to \infty } {a_n} = L\).
(a) If \({a_1} = 1\) and
\({a_{n + 1}} = 1 + \frac{1}{{1 + {a_n}}}\)
Find the first eight terms of the sequence {\({a_n}\)}. Then use part(a) to show that \(\mathop {\lim }\limits_{n \to \infty } {a_n} = \sqrt 2 \). This gives the continued fraction expansion
\(\sqrt 2 = 1 + \frac{1}{{2 + \frac{1}{{2 + ...}}}}\)
After injection of a dose D of insulin, the concentration of insulin in a patient's system decays exponentially and so it can be written as \(D{e^{ - at}}\), where t represents time in hours and a is a positive constant.
(a) If a dose \(D\) is injected every \(T\) hours, write an expression for the sum of the residual concentrations just before the \((n + 1)\)st injection.
(b) Determine the limiting pre-injection concentration.
(c) If the concentration of insulin must always remain at or above a critical value \(C\), determine a minimal dosage \(D\) in terms of \(C\) , \(a\), and \(T\).
\(\sum\limits_{n = 1}^\infty {\cos (\frac{1}{n}} )\) Find Whether It Is Convergent (Or) Divergent. If It Is Convergent Find Its Sum.
Determine whether the geometric series is convergent or divergent..If it is convergent,find its sum.
\(\sum\limits_{n = 1}^\infty {\frac{{{{( - 3)}^{n - 1}}}}{{{4^n}}}} \)
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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