Calculus II, Spring 2016

Midterm 1

Mar 28, 2016 Show all details.

1. (16 pts) Are the integrals ∫_{0}^{1} \frac{1}{\sqrt{x + x^{3}}} dx and ∫_{1}^{∞} tan(\frac{1}{x})dx convergent? Explain.

2. (6 pts) Give formal definition of lim_{n→∞} a_{n} = L.

3. (8 pts) Evaluate lim_{n→∞} (\frac{n − 1}{n + 1})^{n}.

4. (6+10 pts) Show that ∑_{k=1}^{∞} k^{−2} converges and evaluate^{∗} lim_{n→∞} \frac{log ( ∑_{k=n}^{∞} k^{−2})}{log n}. Give details. (If the limit is p, this means that ∑_{k=n}^{∞} k^{−2} is approximately n^{p}. Find p and prove it.) Hint: recall the proof of one of the convergence tests.

5. (8 pts) State (need not prove) Taylor’s Theorem (or Taylor’s formula). Assume the function f has derivatives of all orders on ℝ. Write down n terms approximation and the formula of the remainder term.

6. (6+12 pts)

(a) Show that the series 1 − \frac{1}{2·1!} + \frac{1}{4·2!} − ⋯ + (−1)^{n} \frac{1}{2^{n}·n!} + ⋯ converges absolutely.

(b) Find the sum of the series in (a). \underline{Prove your answer} (that is, the equality holds).

7. (8 pts) Find ∑_{n=1}^{∞} nx^{n} and ∑_{n=1}^{∞} n^{2}x^{n} on |x| < 1 using computational rules of power series. Need NOT prove your equality holds as in previous question.

8. (6 pts) Find a power series that converges on [1, 5) and diverges elsewhere. Explain.

9. (12 pts) Give an approximation of ∫0^{\frac{1}{2}} sin(x^{2})dx to within 10^{−8}. Give the formula of the approximation, but need not find the numerical value. Explain why the error is less than 10^{−8}.

10. (8+10+10 pts) True or False? Prove it if true, give a counter example if false. (a) If ∑ a_{n} converges, then ∑ na_{n} converges.

(b) ^{∗} If ∑ a_{n}x^{n} converges on |x| < 1, then ∑ \sqrt{n}a_{n}x^{n} also converges on |x| < 1.

(c) If g(x) = f(0) + ∑_{n=1}^{∞} \frac{f^{(n)}(0)}{n!}x^{n} converges on |x| < 1, then f(x) = g(x) on |x| < 1.