Chapter 8: Q7E (page 443)
Determine whether the geometric series is convergent or divergent..If it is convergent,find its sum.
\(10 - 2 + .4 - 0.08 + .......\)
The given series is convergent and Sum= \(\frac{{50}}{6}\).
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Suppose you know that\(\left\{ {{{\bf{a}}_{\bf{n}}}} \right\}\)is a decreasing sequence and all its terms lie between the numbers 5 and 8 . Explain why the sequence has a limit. What can you say about the value of the limit?
Determine whether the sequence converges or diverges. If it converges, find the limit.
\({a_n} = \frac{{{{\cos }^2}n}}{{{2^n}}}\)
\(\sum\limits_{n = 1}^\infty {\arctan (n)} \) Find Whether It Is Convergent Or Divergent And Find Its Sum If It Is Convergent.
Let S be the sum of a series of \(\sum {{a_n}} \) that has shown to be convergent by the Integral Test and let f(x) be the function in that test. The remainder after n terms is
\({R_n} = S - {S_n} = {a_{n + 1}} + {a_{n + 2}} + {a_{n + 3}} + .......\)
Thus, Rn is the error made when Sn , the sum of the first n terms is used as an approximation to the total sum S.
(a) By comparing areas in a diagram like figures 3 and 4 (but with x ≥ n), show that
\(\int\limits_{n + 1}^\infty {f(x)dx \le {R_n} \le \int\limits_n^\infty {f(x)dx} } \)
(b) Deduce from part (a) that
\({S_n} + \int\limits_{n + 1}^\infty {f(x)dx \le S \le {S_n} + \int\limits_n^\infty {f(x)dx} } \)
\(\frac{1}{3} + \frac{1}{6} + \frac{1}{9} + \frac{1}{{12}} + \frac{1}{{15}} + ......\)
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