Practice Final Examination 1 – Math 142B
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Practice Final Examination 1 – Math 142B
Question 1. Determine whether each of the following statements is true or false. No justification is required.
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(a) |
If 对 |an | is a convergent series, then 对 an is a convergent series. |
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(b) |
If 对 an and对 bn are convergent, then对 an bn is a convergent series. |
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(c) |
The series 对(−1)n an is convergent if and only if limn→∞ an = 0. |
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(d) |
If 对 anxn and 对 bnxn have radius of convergence r > 0, then 对(an bn )xn has radius of convergence r2 . |
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(e) |
The series 对 xn /(xn + 2) converges uniformly on (−1, 1). |
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(f) |
The Taylor series for e −x2 about x = 0 is 对(−1)nx2n/(2n)!. |
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(g) |
If the Taylor series T(x) for f(x) about x = 0 satisfies T(1) = f(1), then T(−1) = f(−1). |
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(h) |
If a function on [0, 1] is bounded, then it is Riemann integrable. |
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(i) |
If a function on [0, 1] is Riemann integrable, then it is bounded. |
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(j) |
If f,g : [0, 1] → R are integrable, then so is their product f · g . |
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Question 2. Let an = 
for n ≥ 1. For each part of this question, justify your answers.
(a) Prove that 对 an diverges.
(b) Prove that 对(−1)n an converges.
(c) Prove that 对 an(2) converges.
Question 3.
For n ≥ 1 and x ∈ [0, 1], let
fn (x) =
and
(a) Prove that for any T ∈ (0, 1), 对 fn (x) converges uniformly to f(x) on the interval [−T,T].
(b) Prove that f is continuous on (−1, 1).
(c) Prove that 对 fn (x) does not converge uniformly to f(x) on the in- terval (−1, 1).
(d) Does 对 fn (x) converge pointwise to f(x) for x ∈ [−1, 1)?
Question 4.
Let a ≥ 0 and let fn (x) = nx/(nxa +1) for n ≥ 1 and x ∈ [0, 1]. Justify your answers for each part of this question.
(a) Find a function f : [0, 1] → R such that fn → f pointwise on [0, 1]. (b) For which values of a is f bounded on [0, 1]?
(c) For which values of a is the convergence fn → f uniform on [0, 1]?
(d) When the limit exists, determine
n(l)
\0 1 fn (x)dx.
2022-09-01