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Convergence and Divergence of Sequences and Series: A Comprehensive Quiz

Test your knowledge of sequence and series convergence with this engaging quiz! Explore important concepts such as the convergence criteria for sequences, the Taylor series expansion, and specific series approximations. Challenge yourself with questions covering sequences like an = n+3/n and series like arctan(x) for approximating π/4. Determine which series diverge, explore Taylor series coefficients, and practice evaluating convergence criteria. Perfect for students and math enthusiasts looking to sharpen their skills.

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Convergence and Divergence of Sequences and Series: A Comprehensive Quiz

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  1. Quiz!!!

  2. Which sequence converges? • an = n+3/n • an = -1+(-1)^n/n • an = n/lnn • an = n!/3^n Answer: B

  3. 2. Which of the following sequences diverges? A. an = 1/n B. an = 2^n/e^n C. an = n^2/e^n D. an = n/lnn Answer: D

  4. 3. The sequence {r^n} converges if and only if • IrI<1 • IrI<=1 • -1<r<=1 • 0<r<1 Answer: C

  5. 4. If the series tan^(-1)=1-1/3+1/5-1/7+… is used to approximate Π/4 with error less than 0.001, then the smallest number of terms needed is • 100 • 200 • 400 • 500 Answer: D

  6. 5. The coefficient of (x-Π/4)^3 in Taylor series about Π/4 of f(x)=cosx is A.(√3)/12 B. -1/12 C. 1/12 D. 1/ (6√2) Answer: D

  7. 6. Which of the following series can be used to compute ln0.8? • ln(x-1) expanded around x=0 • lnx about x=0 • lnx expanded about x=1 • none of these Answer: C

  8. 7. The sum of the series ∞ ∑(Π^3/3^Π)^n is equal to n=1 • 0 B. 1 C. Π^3/(3^Π-Π^3) D. none of these Answer: C

  9. 8. The set of all values of x for which ∞ ∑n*2^n/x^n converges is n=1 • only x=0 B. IxI=2 C. IxI>2 D. none of these Answer: C.

  10. 9. The coefficient of (x-1)^5 in the Taylor series for xlnx about x=1 is • -1/20 • 1/5! • -1/5! • 1/4! Answer: A.

  11. 10. The coefficient of x^2 in the Maclaurin series or e^(sinx) is • 0 • 1 • 1/2! • -1 Answer: C

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