Chapter 8: Q11RE (page 497)
Determine whether the series is convergent or divergent.
\(\sum\limits_{{\rm{n = 1}}}^\infty {\frac{{{{\rm{n}}^{\rm{3}}}}}{{{{\rm{5}}^{\rm{n}}}}}} \)
The series is perfectly convergent.
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A certain ball has the property that each time it falls from a height \(h\)\(\) onto a hard, level surface, it rebounds to a height \(rh\), where \(0 < r < 1\). Suppose that the ball is dropped from an initial height of \(H\) meters.
(a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels.
(b) Calculate the total time that the ball travels. (Use the fact that the ball falls \(\frac{1}{2}g{t^2}\) meters in \({t^{}}\)seconds.)
(c) Suppose that each time the ball strikes the surface with velocity \(v\) it rebounds with velocity \( - kv\) , where \(0 < k < 1\). How long will it take for the ball to come to rest?
Use definition 2 directly to prove that \(\mathop {\lim }\limits_{n \to \infty } {(0.8)^n} = 0\)(from ? with, \(r = 0.8\)). Use logarithms to determine how large n has to be so that \((0.8) < 0.000001\).
We have seen that the harmonic series is a divergent series whose terms approach 0. Show that \(\sum\limits_{n = 1}^\infty {\ln (1 + \frac{1}{n})} \) is another series with this property.
Determine whether the series is convergent or divergent. If its convergent, find its sum.
\(\sum\limits_{n = 1}^\infty {\frac{{{3^n}}}{{{e^{(n - 1)}}}}} \)
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
\({S_n} + \int\limits_{n + 1}^\infty {f(x)dx \le S \le {S_n} + \int\limits_n^\infty {f(x)dx} } \)
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