## Transcribed Text

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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