 # 1. For the sequence (xn) with X1 = 7 and xn+1 = √ , for n 1 ...

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1. For the sequence (xn) with X1 = 7 and xn+1 = √ , for n 1 (a) (5 pts) Show (xn) converges. (b) ( 5 pts) Find . Be sure to show your work. 2. (10 pts) The sequence (an) satisfies | an+1 - an | < . Prove that (an) is a Cauchy sequence. 3. (10 pts) Find the subsequential limits of the sequence an = (-1)n sin ( ) 4. (10 pts) Determine whether the series ∑ converges or diverges. Explain your reasoning. 5. (10 pts) Determine whether the series ∑ converges or diverges. Explain your reasoning. 6. (10 pts) Let f(x) be a function with f' (x) defined for all x ∈ R. Show that if f'(x) is bounded, that is, |f' (x)| M for some M > 0 and for all x ∈ R, then f(x) is uniformly continuous on R. 7. (10 pts) Prove that the series ∑ ⁄ converges uniformly on any bounded interval [0, M] for some M > 0, but does NOT converge uniformly on [0, ). (Hint: Use the M-test and the Cauchy criterion.) 8. Let f(x) be a continuous function on [a, b] (- ). Suppose F(x) is a differentiable function on (a,b) such that F' (x) = f(x). (a) (5 pts) Find the limit [ ( ) ( ) ( )] Justify why the limit exists and your answer. (b) (5 pts) Compute the limit ( ) 9. Consider the power series ∑ (a) (5 pts) For what values of x does the series converge? What is the pointwise limit function? (Hint: use the Taylor series of (b) (5 pts) Give an interval of x where the series converges uniformly. Explain why? (c) ( 5 pts) On the interval from your answer to (b), does the following series ∑ converge? Explain why. If yes, what function does it converge to? 10. Let f be the function defined on [-π, π]] by ∫ (a) (5 pts) Show that f is differentiable on (-π, π ). What is its derivative function? (b) (5 pts) Evaluate f (0) and f'(√ /2)

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