The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $ \(\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}\right\rbrace}}}{{{a}_{{{n}}}}}}{<}\frac{{1}}{{2}}\) then \(\displaystyle\sum{a}_{{{n}}}\) converges, while if \(\displaystyle\lim{\left\lbrace{n}\rightarrow\infty\right\rbrace}{\frac{{{a}{\left\lbrace{2}{n}+{1}\right\rbrace}}}{{{a}_{{{n}}}}}}{>}\frac{{1}}{{2}}\) then \(\displaystyle\sum{a}_{{{n}}}\) $ diverges. Let \(\displaystyle{a}_{{{n}}}={\frac{{{n}^{{{\ln{{n}}}}}}}{{{\left({\ln{{n}}}\right)}^{{{n}}}}}}\). Show that \(\frac{a_{2n}}{a_{n}} \rightarrow 0\) as \(\displaystyle{n}\rightarrow\infty\)
B. G. Cosmos, a scientist, believes that the probability is \(\displaystyle{\frac{{{2}}}{{{5}}}}\) that aliens from an advanced civilization on Planet X are trying to communicate with us by sending high-frequency signals to Earth. By using sophisticated equipment, Cosmos hopes to pick up these signals. The manufacturer of the equipment, Trekee, Inc., claims that if aliens are indeed sending signals, the probability that the equipment will detect them is \(\displaystyle{\frac{{{3}}}{{{5}}}}\). However, if aliens are not sending signals, the probability that the equipment will seem to detect such signals is \(\frac{1}{10}\). If the equipment detects signals, what is the probability that aliens are actually sending them?
The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence. The test states that if $ \(\lim_{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\) then \(\displaystyle\sum{a}_{{{n}}}\) converges, while if \(\lim_{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\) then \(\displaystyle\sum{a}_{{{n}}}\) diverges. Let \(\displaystyle{a}_{{{n}}}={\frac{{{n}^{{{\ln{{n}}}}}}}{{{\left({\ln{{n}}}\right)}^{{{n}}}}}}\). Show that \(\frac{a_{2n}}{a_{n}} \rightarrow 0\) as \(\displaystyle{n}\rightarrow\infty\)
Driven by technological advances and financial pressures, the number of surgeries performed in physicians' offices nationwide has been increasing over the years. The function \(\displaystyle{f{{\left({t}\right)}}}=-{0.00447}{t}^{{{3}}}+{0.09864}{t}^{{{2}}}+{0.05192}{t}+{0.8}{\left({0}\leq{t}\leq{15}\right)}\) gives the number of surgeries (in millions) performed in physicians' offices in year t, with \(t=0\) corresponding to the beginning of 1986.
a. Plot the graph of f in the viewing window \([0,15]\times [0,10]\).
b. Prove that f is increasing on the interval [0, 15].
The following advanced exercise use a generalized ratio test to determine convergence of some series that arise in particular applications, including the ratio and root test, are not powerful enough to determine their convergence.
The test states that if
\(\lim_{n \rightarrow \infty} \frac{a_{2 n}}{a_{n}}<1 / 2\)
then \(\sum a_{n}\) converges,while if
\(\lim_{n \rightarrow \infty} \frac{a_{2 n+1}}{a_{n}}>1 / 2\),
then \(\sum a_{n}\) diverges. Let \(\displaystyle{a}_{{{n}}}={\frac{{{1}}}{{{1}+{x}}}}{\frac{{{2}}}{{{2}+{x}}}}\ldots{\frac{{{n}}}{{{n}+{x}}}}{\frac{{{1}}}{{{n}}}}={\frac{{{\left({n}-{1}\right)}!}}{{{\left({1}+{x}\right)}{\left({2}+{x}\right)}\ldots{\left({n}+{x}\right)}}}}\).
Show that \(a_{2 n}/a_{n} \leq e^{-x/2}/ 2\) .
For which x > 0 does the generalized ratio test imply convergence of \(\displaystyle{\sum_{{{n}={1}}}^{\infty}}{a}_{{{n}}}\)?
