EE 518 : Homework #4 Due on Monday 2:00pm, September 26, 2016Problem 1Let an=αnn!. Please show that the sequence {an} is convergent, and limn→∞αnn!= 0, where α ∈ R is any realnumber.Problem 2Find the limits of the following sequences.(i)√2,p2√2,q2p2√2, . . .(ii) an= (√n + 1 −√n)(iii) an=n sin n!n2+1Problem 3Let {xn} and {yn} be two number sequences, suppose that for some fixed integer K, we haveyn≥ xn≥ 0, ∀n ≥ K.Please check whether following statements are true or false. If true, prove them. If false, provide at leastone counter-example.(i) The convergence of {yn} implies the convergence of {xn};(ii) The divergence of {xn} implies the divergence of {yn};(iii) The convergence ofP∞n=0xnimplies the convergence ofP∞n=0yn;(iv) The divergence ofP∞n=0xnimplies the divergence ofP∞n=0yn.Problem 4Discuss convergence or divergence for each of the following series:(i)P∞n=1n2n;(ii)P∞n=1n sin1n;(iii)P∞n=112n+lnn33n(iv)P∞n=1n3n+1an, for a > 0Problem 5Let f(x) =1 + ln(x + ex)sin xx1x2+4and g(x) =hx3+ 1 5exx2cos x−1iFind the following limits:(i) limx→0f(x), limx→0g(x)(ii) limx→0(g · f)(x)Page 1 of 2EE 518 : Homework #4 Due on Monday 2:00pm, September 26, 2016Problem 6Find the continuous intervals and discontinuities of following functions. Indicate what kind of discontinuitiesthey are.(i) f(x) =x2− x|x|(x2− 1)(ii) f(x) =sin xx− ex+14, x < 012, x = 0√x + 4 − 2x, x > 0Problem 7Let Sn=nXk=11k2and write a MATLAB function that finds the minimum required n for Snto reach the sumlimitπ26within a given positive error that is less than 1. The input argument of the function should beerror and the output should be n, such that:Sn−π26≤ error.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