EE 518 Homework 4 Due on Monday 2 00pm September 26 2016 Problem 1 n n Let an Please show that the sequence an is convergent and lim 0 where R is any real n n n number Problem 2 Find the limits of the following sequences q p p i 2 2 2 2 2 2 ii an n 1 n iii an n sin n n2 1 Problem 3 Let xn and yn be two number sequences suppose that for some fixed integer K we have yn xn 0 n K Please check whether following statements are true or false If true prove them If false provide at least one counter example i The convergence of yn implies the convergence of xn ii The divergence of xn implies the divergence of yn P P iii The convergence of n 0 xn implies the convergence of n 0 yn P P iv The divergence of n 0 xn implies the divergence of n 0 yn Problem 4 Discuss convergence or divergence for each of the following series P n i n 1 2n P 1 ii n 1 n sin n P 1 lnn 3 iii n 1 2n 3n P n an iv for a 0 n 1 3n 1 Problem 5 1 ln x ex h sin x and g x x3 1 Let f x x 1 5ex x2 cos x 1 i Find the following limits x2 4 i lim f x lim g x x 0 x 0 ii lim g f x x 0 Page 1 of 2 EE 518 Homework 4 Due on Monday 2 00pm September 26 2016 Problem 6 Find the continuous intervals and discontinuities of following functions Indicate what kind of discontinuities they are x2 x x x2 1 sin x 1 ex x 4 1 ii f x 2 x 4 2 x i f x x 0 x 0 x 0 Problem 7 Let Sn n X 1 and write a MATLAB function that finds the minimum required n for Sn to reach the sum k2 k 1 2 limit within a given positive error that is less than 1 The input argument of the function should be 6 error and the output should be n such that Sn 2 error 6 With your function find out the values of n when error is 0 1 0 05 0 01 0 001 and 0 0001 respectively Hint Write a function that computes Sn then write the function calling this function to find n Page 2 of 2