18. Sequences

Homework

1. Consider the sequence, \(\left\{\dfrac{0}{2}, \dfrac{3}{5}, \dfrac{8}{10}, \dfrac{15}{17}, \dfrac{24}{26}, \dfrac{36}{37}, \cdots\right\}\). Assume the pattern continues. Write out the general term of the sequence, \(a_n\). Assume the terms are numbered beginning with \(n=1\).

2. For the sequence, \(a_n=\dfrac{4\cdot2^n+9\cdot3^n}{2^{n+1}+3^{n+1}}\), use a computer algebra system (Maple, Desmos, Geogebra, etc. Say which one.) to plot the first \(10\) terms and guess the limit of the sequence.

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In each problem, determine if the sequence is convergent or divergent, If it converges, find the limit. If it diverges, determine if it diverges to plus infinity or minus infinity or is oscillatory divergent. Be sure to explain the method you use.

3. \(a_n=\dfrac{2n^6-6n^7}{4n^4-3n^5}\)

4. \(a_n=\dfrac{4\cdot2^n+9\cdot3^n}{2^{n+1}+3^{n+1}}\)
   NOTE: This is the same sequence as in problem (2).

5. \(a_n=\dfrac{6n^4+(-1)^nn^3+7}{2n^4-4n^3}\)
   NOTE: You cannot use l'Hopital's Rule because the numerator is not differentiable.

6. \(a_n=\dfrac{2+3n^2+(-1)^n4n^3}{n^2-2n^3}\)
   NOTE: You cannot use l'Hopital's Rule because the numerator is not differentiable.

7. \(b_n=\arcsin \sqrt{\dfrac{2n^2+1}{4n^2}}\)

8. \(p_n=\dfrac{n^2}{n-1}-\dfrac{n^3}{n^2-1}\)

9. \(q_n=\sqrt{4n+\sqrt{n}}-\sqrt{4n}\)

10. \(a_n=\left(1-\dfrac{4}{n}\right)^{3n}\)

11. \(a_n=\left(1-\dfrac{4}{\sqrt{n}}\right)^{3n}\)

12. \(a_n=\dfrac{e^n}{n^e}\)   (Honors Only)
    HINT: What happens when an exponent is negative?

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In each problem, first assume the limit exists and find the possible limit values. Next, write out the first \(4\) terms of the sequence, exactly and as decimals. You may use a calculator or computer to compute the terms. Make a conjecture saying whether the sequence is increasing or decreasing and bounded above or below by what number. Then use the Bounded, Monotonic Sequence Theorem to determine whether the sequence converges or diverges. In the process, use Mathematical Induction to prove monotonicity and boundedness. If it converges, find the limit. If it diverges, determine if it diverges to \(\infty\) or \(-\infty\) or is oscillatory divergent.
Maple Code to find terms.

Maple Code to compute the terms of a recursive seqence.

Suppose you have the recursive sequence \[ a_{n+1}=\dfrac{ {a_n}^2+9}{2a_n} \quad \text{with} \quad a_1=2 \] This code will compute the first \(4\) terms and their decimal approximations:
  f:=a-> (a^2+9)/(2*a);
  a1:=2;
  a2:=f(a1); evalf(%);
  a3:=f(a2); evalf(%);
  a4:=f(a3); evalf(%);

13. \(a_1=4,\qquad a_{n+1}=\sqrt{10a_n-16}\)

14. \(a_1=104,\qquad a_{n+1}=\sqrt{10a_n-16}\)

15. \(a_1=8,\qquad a_{n+1}=\dfrac{1}{4} {a_n}^2\)

16. \(a_1=2,\qquad a_{n+1}=\dfrac{1}{4} {a_n}^2\)

17. \(a_1=4,\qquad a_{n+1}=4\sqrt{a_n}\)

18. \(a_1=64,\qquad a_{n+1}=4\sqrt{a_n}\)

19. \(a_1=\dfrac{1}{3},\qquad a_{n+1}=\dfrac{1}{1-a_n}\)   (Honors Only)