Let fnx 12cos2nxSquarerootn Prove carefully that fn converg

Let f_n(x) = 1+2cos^2nx/Squarerootn. Prove carefully that (f_n) converges uniformly to 0 on R. 24.2 For x [0, infinity), let f_n(x) = x/n. (a) Find f(x) = lim f_n(x). (b) Determine whether f_n rightarrow f uniformly on [0,1]. (c) Determine whether f_n rightarrow f uniformly on [0, infinity). 24.3 Repeat Exercise 24.2 for f_n(x) = 1/1+x^n. 24.4 Repeat Exercise 24.2 for f_n(x) = x^n/1+x^n. 24.5 Repeat Exercise 24.2 for f_n(x) = x^n/n+x^n. 24.6 Let f_n(x) = (x - 1/n)^2 for x [0, 1]. (a) Does the sequence (f_n) converge pointwise on the set [0,1]? If so, give the limit function. (b) Does (f_n) converge uniformly on [0,1]? Prove your assertion. 24.7 Repeat Exercise 24.6 for f_n(x) = x - x^n.

Solution

(24.1) We have -1<cosnx <1

Now 0< cos²nx < 1

=> 0 < 2cos²nx < 2

=> 0+1 < 1+2cos²nx < 3

=> 1/sqrtn <( 1+2cos²nx )/sqrtn < 3/sqrtn

Now as n approaches infinity, 1/sqrtn _> 0 and also 3/sqrtn -> 0

Hence by squueze principle as both the functions on either side approaches 0 so the function in middle of these must approach 0 as n approaches infinity

Hence (1+2cos²nx)/sqrtn converges uniformly to 0 on R