# Mathematical Food for Thought

Serves a Daily Special and an All-You-Can-Eat Course in Problem Solving. Courtesy of me, Jeffrey Wang.

• ## Meta

 To Converge Or Not To Converge. Topic: Real Analysis/S&S. November 27th, 2006 Problem: (Stanford Math 51H, Cauchy Root Test) Suppose there exists a and an integer such that for all . Prove that converges. Solution: Well let’s just discard because it is finite and obviously converges. It remains to show that converges. But then we have . So . The last summation, however, is a geometric series with common ratio , so it converges. Hence our sum does as well. QED. ——————– Comment: The root test is, in effect, a comparison to a geometric series. The hypothesis is that we can bound by a geometric series for all large , implying convergence. ——————– Practice Problem: (Stanford Math 51H) Discuss the convergence of . Posted in Real Analysis, Sequences & Series || 3 Comments » Limitten. Topic: Real Analysis. November 23rd, 2006 Definition: A sequence has limit , where is a given real number, if for each there is an integer such that for all integers . ——————– Problem: (Stanford Math 51H) Prove that a sequence cannot have more than one limit. Solution: This is logically true, as usual, but a rigorous argument is much more fun. Suppose and . Letting , we know there exists such that for for . So the above two inequalities are true for . Adding the two inequalities together, we get . However, by the triangle inequality we know (by setting is the usual one), so . Then , a contradiction. Hence can have at most one limit. QED. ——————– Comment: The real definition of the limit is almost always complete overlooked in a regular calculus course (i.e. Calc AB and BC). But it’s pretty much the foundation of all of calculus so it is really quite nice to know and work with. ——————– Practice Problem: (Stanford Math 51H, Sandwich Theorem) If are given convergent sequences with , and if is any sequence such that , prove that is convergent and . Posted in Real Analysis || No Comments » Root Beer. Topic: Real Analysis. November 22nd, 2006 Problem: (Stanford Math 51H) Prove that every positive real number has a positive square root (That is, for any , prove that there is a real number such that ). [Usual properties of the integers are assumed.] Solution: Consider the set . We can show that is non-empty by selecting an integer large enough such that . Since is a real and the integers are unbounded, there exists a positive integer such that , thus so is bounded from above. Now there must exist such that . We claim that . Suppose the contrary; then either or . CASE 1: If , then consider . Choose large enough such that which is definitely true for any . But then and , contradicting the fact that is the supremum. CASE 2: If , then consider . Again, choose large enough such that which we know is true for (furthermore, we impose the restriction so our resulting real is positive). Then for all and , contradicting the fact that is the supremum. Hence we know that , or that the square root of any positive real number exists and is a positive real number. QED. ——————– Comment: This is from the real analysis portion of a freshman honors calculus course, i.e. a rigorous treatment of the real numbers which is the basis for calculus like limits and stuff. Really understanding calculus involves really understanding how the real number system works. ——————– Practice Problem: (Stanford Math 51H) If with , prove: (a) There is a rational . (b) There is an irrational . (c) contains infinitely many rationals and infinitely many irrationals. Posted in Real Analysis || 4 Comments »