Problem 45
For \(k \geq 0\), find $$ \lim _{n \rightarrow \infty} \frac{1^{k}+2^{k}+\cdots+n^{k}}{n^{k+1}} $$ Hint: Though this has the \(\infty / \infty\) form, l'Hôpital's Rule is not helpful. Think of a Riemann sum.
Problem 46
Let \(c_{1}, c_{2}, \ldots, c_{n}\) be positive constants with \(\sum_{i=1}^{n} c_{i}=1\), and let \(x_{1}, x_{2}, \ldots, x_{n}\) be positive numbers. Take natural logarithms and then use l'Hôpital's Rule to show that $$ \lim _{t \rightarrow 0^{+}}\left(\sum_{i=1}^{n} c_{i} x_{i}^{t}\right)^{1 / t}=x_{1}^{c_{1}} x_{2}^{c_{2}} \cdots x_{n}^{c_{n}}=\prod_{i=1}^{n} x_{i}^{c_{i}} $$ Here \(\prod\) means product; that is, \(\prod_{i=1}^{n} a_{i}\) means \(a_{1} \cdot a_{2} \cdots \cdots a_{n} .\) In particular, if \(a, b, x\), and \(y\) are positive and \(a+b=1\), then $$ \lim _{t \rightarrow 0^{+}}\left(a x^{t}+b y^{t}\right)^{1 / t}=x^{a} y^{b} $$
Problem 47
Use the Comparison Test of Problem 46 to show that \(\int_{1}^{\infty} e^{-x^{2}} d x\) converges. Hint: \(e^{-x^{2}} \leq e^{-x}\) on \([1, \infty)\)
Problem 48
Consider \(f(x)=n^{2} x e^{-n x}\). (a) Graph \(f(x)\) for \(n=1,2,3,4,5,6\) on \([0,1]\) in the same graph window. (b) For \(x>0\), find \(\lim _{n \rightarrow \infty} f(x)\). (c) Evaluate \(\int_{0}^{1} f(x) d x\) for \(n=1,2,3,4,5,6\). (d) Guess at \(\lim _{n \rightarrow \infty} \int_{0}^{1} f(x) d x\). Then justify your answer rigorously.
Problem 49
Find the absolute maximum and minimum points (if they exist) for \(f(x)=\left(x^{25}+x^{3}+2^{x}\right) e^{-x}\) on \([0, \infty)\).
Problem 53
(Gamma Function) Let \(\Gamma(n)=\int_{0}^{\infty} x^{n-1} e^{-x} d x, n>0 .\) This integral converges by Problems 51 and \(52 .\) Show each of the following (note that the gamma function is defined for every positive real number \(n\) ): (a) \(\Gamma(1)=1\) (b) \(\Gamma(n+1)=n \Gamma(n)\) (c) \(\Gamma(n+1)=n !\), if \(n\) is a positive integer.
Problem 54
Evaluate \(\int_{0}^{\infty} x^{n-1} e^{-x} d x\) for \(n=1,2,3,4\), and 5, thereby confirming Problem \(53(\mathrm{c})\).
Problem 57
By interpreting each of the following integrals as an area and then calculating this area by a \(y\) -integration, evaluate: (a) \(\int_{0}^{1} \sqrt{\frac{1-x}{x}} d x\) (b) \(\int_{-1}^{1} \sqrt{\frac{1+x}{1-x}} d x\)
Problem 58
Suppose that \(0
