Math 312 Intro to Real Analysis Midterm Exam 2 Stephen G Simpson Friday March 27 2009 1 True or False 2 points each a Every monotone sequence of real numbers is convergent b Every sequence of real numbers has a lim sup and a lim inf c Every sequence of real numbers has a monotone subsequence d Every sequence of real numbers has a convergent subsequence e If lim inf an lim sup an then lim an f We can find a sequence of real numbers an such that the subsequential limits of an are exactly the real numbers in the closed interval 1 1 1 1 1 is absolutely convergent g The series 1 4 9 16 P P h If an is convergent then so is an 2 P P 2 i If an is convergent then so is an X 1 j is convergent for all p 1 np n 1 k X 1 1 n 10 9 n 1 l The function x is uniformly continuous on 0 m If a function is uniformly continuous on the interval a b and on the interval b c then it is uniformly continuous on the interval a c 2 6 points each Which of the following series are convergent and or absolutely convergent Please indicate which tests you are using and show your work 1 1 1 2 3 4 p X b n2 n n a 1 c n 1 X p n2 1 n n 1 1 1 01n 1000 X 1 e 2 n n2 1 X n 1 f log n n 1 d X 1 3 8 points Use algebra plus limit laws to calculate log e 2n 11 n lim sin 4n 5 16n In performing this calculation you may take it for granted that functions such as sin x cos x ex log x x etc are continuous Please show your work 4 5 points Assume that f x is continuous on the closed interval 0 1 Assume also that f x 0 1 for all x 0 1 Using known theorems about continuous functions prove that the equation f x x has at least one solution in 0 1 5 3 points each Which of the following functions are continous and or uniformly continuous on the specified domain a 1 x on its natural domain b 1 x on 1 c d e 1 x cos on 0 10 x x on 0 x x for x 6 0 f x 0 for x 0 on 6 10 points It is known that the function x is uniformly continuous on the interval 0 01 100 Given 0 find a 0 depending only on such that x y whenever 0 01 x y 100 and x y 2