Chapter 8: Q21E (page 463)
Determine whether the series is absolutely convergent, conditionally convergent,or divergent.
\(\sum\limits_{n = 0}^\infty {\frac{{{{( - 10)}^n}}}{{n!}}} \)
The Series is Divergent.
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Graph the curves\(y = {x^n}\),\(0 \le x \le 1\), for\(n = 0,1,2,3,4,....\)on a common screen. By finding the areas between successive curves, give a geometric demonstration of the fact, shown in Example 6, that
\(\sum\limits_{n = 1}^\infty {\frac{1}{{n(n + 1)}}} = 1\)
What can you say about the series \(\sum {{a_n}} \) in each of the following cases?
(a) \(\mathop {lim}\limits_{n \to \infty } \left| {\frac{{{a_{n + 1}}}}{{{a_n}}}} \right| = 8\)
(b) \(\mathop {lim}\limits_{n \to \infty } \left| {\frac{{{a_{n + 1}}}}{{{a_n}}}} \right| = 0.8\)
(c) \(\mathop {lim}\limits_{n \to \infty } \left| {\frac{{{a_{n + 1}}}}{{{a_n}}}} \right| = 1\)
Determine whether the sequence converges or diverges. If it converges, find the limit.
\({a_n} = \frac{{{{( - 1)}^n}}}{{2\sqrt n }}\)
Determine whether the sequence converges or diverges. If it converges, find the limit.
\({a_n} = \frac{{{{\cos }^2}n}}{{{2^n}}}\)
Express the number as a ratio of integers.
\(\)\(0.\overline {46} = 0.46464646...\)
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