When working on this assignment, please follow the guidelines listed in the prev

Question

When working on this assignment, please follow the guidelines listed in the previous assignments. Find the limits of given sequences Tou may use any of the theorems form Section 2.2 xn = n + 5 - n yn = n2 + 5 - n2 + 1/ n + 2 - n Let (xn} be a sequence and r (0, 1). Suppose that for every n N we have |xn - xn + 1| rn. Prove that the sequence converges. (Hint: imitate the argument used in one of the class examples.) Prove, directly from the definition, that the following limits exist. lim x2 - 5x + 4 = -2 lim 3x2 + 2x + 1 = 2 lim x3 - 5 = 3 Evaluate the following limits using the limit theorems, comparison theorem, and squeeze theorem of Section 1.1(but do not use Hospital's rule).You may assume

Explanation / Answer

Now for every n belonging to N , that is n >= 1 , r^n is always less than 1 ; because for any no. less than 1 , raising their power decreases the value. For example 0.1^2 = 0.01 etc. hence |Xn - Xn+1| , the max value is r^n which is less than 1 and keeps moving towards zero Hence the difference between 2 adjacent terms of series approaches zero as the value of n increases. Hence the series is convergent

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