Chapter 8: Q36RE (page 498)
Find the radius of convergence and interval of convergence of the series.
\(\sum\limits_{{\rm{n = 1}}}^\infty {{{{\rm{( - 1)}}}^{\rm{n}}}} \frac{{{{\rm{x}}^{\rm{n}}}}}{{{{\rm{n}}^{\rm{2}}}{{\rm{5}}^{\rm{n}}}}}\)
The Interval of convergence is\({\rm{[ - 5,5]}}\).
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If the nth partial sum of a series\(\sum\limits_{n = 1}^\infty {{a_n}} \)is\({s_n} = \frac{{n - 1}}{{n + 1}}\)find\({a_n}\)and\(\sum\limits_{n = 1}^\infty {{a_n}} \).
Determine whether the series is convergent or divergent:\(\sum\limits_{n = 1}^\infty {\frac{{n + 5}}{{\sqrt(3){{{n^7} + {n^2}}}}}} \).
Determine whether the sequence converges or diverges. If it converges, find the limit.
\({a_n} = \sqrt {\frac{{n + 1}}{{9n + 1}}} \)
Use definition 2 directly to prove that \(\mathop {\lim }\limits_{n \to \infty } {r^n} = 0\)when\(\left| r \right| < 1\)
(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.
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