Problem 85
Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n}}{\sqrt{n}} $$
Problem 86
Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} n !}{3^{n}} $$
Problem 87
Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \frac{(-1)^{n} n !}{n^{n}} $$
Problem 88
Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \sin \left(\frac{n \pi}{2}\right) $$
Problem 89
Is the series convergent or divergent? If convergent, is it absolutely convergent? $$ \sum_{n=1}^{\infty} \cos (\pi n) e^{-n} $$
Problem 90
Evaluate. $$ \sum_{n=1}^{\infty} \frac{2^{n+4}}{7^{n}} $$
Problem 91
Evaluate. $$ \sum_{n=1}^{\infty} \frac{1}{(n+1)(n+2)} $$
Problem 92
A legend from India tells that a mathematician invented chess for a king. The king enjoyed the game so much he allowed the mathematician to demand any payment. The mathematician asked for one grain of rice for the first square on the chessboard, two grains of rice for the second square on the chessboard, and so on. Find an exact expression for the total payment (in grains of rice) requested by the mathematician. Assuming there are 30,000 grains of rice in 1 pound, and 2000 pounds in 1 ton, how many tons of rice did the mathematician attempt to receive?
Problem 93
The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula \(x_{n+1}=b x_{n}\), where \(x_{n}\) is the population of houseflies at generation \(n\), and \(b\) is the average number of offspring per housefly who survive to the next generation. Assume a starting population \(x_{0}\). Find \(\lim _{n \rightarrow \infty} x_{n}\) if \(b>1, b<1\), and \(b=1\).
Problem 94
The following problems consider a simple population model of the housefly, which can be exhibited by the recursive formula \(x_{n+1}=b x_{n}\), where \(x_{n}\) is the population of houseflies at generation \(n\), and \(b\) is the average number of offspring per housefly who survive to the next generation. Assume a starting population \(x_{0}\). Find an expression for \(S_{n}=\sum_{i=0}^{n} x_{i}\) in terms of \(b\) and \(x_{0}\). What does it physically represent?
