Chapter 8: Q13E (page 463)
Approximate the sum of the series correct to four decimal places.
\(\sum\limits_{n = 1}^\infty {\frac{{{{\left( {{\rm{ - }}1} \right)}^n}}}{{\left( {2n} \right){\rm{!}}}}} \)
Sum Upto 4 Decimal Places Is -0.4597
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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
\(\sqrt 2 = 1 + \frac{1}{{2 + \frac{1}{{2 + ...}}}}\)
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\(\sum\limits_{n = 1}^\infty {{a_n}} \).
Show that if we want to approximate the sum of the series\(\sum\limits_{n = 1}^\infty {{n^{ - 1.001}}} \)so that the error is less than\(5\)in the ninth decimal place, then we need to add more than\({10^{11,301}}\)terms!
The meaning of the decimal representation of a number\(0 \cdot {d_1}{d_2}{d_3}\)(where the digit\({d_1}\)is one of the numbers\(0,1,2,........9\)) is that
\(0 \cdot {d_1}{d_2}{d_3}{d_4}...... = \frac{{{d_1}}}{{10}} + \frac{{{d_2}}}{{{{10}^2}}} + \frac{{{d_3}}}{{{{10}^3}}} + \frac{{{d_4}}}{{{{10}^4}}} + ....\)show that this series always converges.
Determine whether the series is convergent or divergent: \(\sum\limits_{n = 1}^\infty {\sin \frac{1}{n}} \).
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