Math Homework

Exercise 1 (5 Points) Justify why the power set, denoted by \mathcal{P}(A), of a finite set A has 2^{|A|} many elements. Here, |A| denotes the number of elements in A.

Exercise 2 (1 Point) Draw the graphs of \log_{e}(x) and \log_{2}(x).

Exercise 3 (2 Point) Identify at least two key differences in the above two plots.

Exercise 4 (2 Points) Draw the graphs of x^2, \sqrt{x}, and x in the same plot.

Exercise 5 (2 Points) Can you find an m such that the line y=mx stays above the graph of \sqrt{x} for any large positive x?

Exercise 6 (2 Points) Can you find an m such that the line y=mx stays above the graph of x^2 for any large positive x?

Exercise 7 (2 Point) Find the value of 1 + 2 + 3 + \ldots + 50 =? Hint: Use the formulas in Appendix

Exercise 8 (2 Point) Find the value of 2 + 4 + 6 + \ldots + 100 =?

Exercise 9 (2 Point) Find the value of \frac{1}{3} + \frac{1}{9} + \frac{1}{27} + \ldots =? Hint: Use the formulas in Appendix

Exercise 10 (Bonus) Find the value of \frac{1}{2} + \frac{1}{3} + \frac{1}{4} + \ldots =?

Appendix

  1. Sum of natural numbers: \sum_{k=1}^n k=\frac{n(n+1)}{2}.

  2. Finite geometric series: \sum_{k=0}^n x^k=\frac{x^{n+1}-1}{x-1}.

  3. Infinite geometric series for |x|<1: \sum_{k=0}^\infty x^k=\frac{1}{1-x}.