Section 11.1: Sequences: \( \{ a_n \}_{n=1}^{\infty} \)

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1. Find a formula for the general term \(a_{n}\) of the sequence, assuming that the pattern of the first few terms continues. (Assume that \(n\) begins with 1.)

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  1. \(\{2,5,8,11,14, \ldots\}\)
  2. \( \{3, -6, 12, -24, \ldots \} \)
  3. \(\{8, -4, 2, -1, \ldots \}\)
  4. \( \left\{ 2, -\dfrac{4}{3}, \dfrac{8}{9},-\dfrac{16}{27}, \dfrac{32}{81}, \ldots \right\}\)

2. Determine if the sequence converges or diverges. If it converges, determine its limit.

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  1. \( \displaystyle a_n= \dfrac{2n^2+2n+10}{3n^3 -n^2-5}\)
  2. \( \displaystyle a_n= \dfrac{2n^4 -3n}{24n^4+ 100n^2+10}\)
  3. \( \displaystyle a_n=\dfrac{3n^3+8n^2-10}{500n^2+20n+30}\)
  4. \( \displaystyle a_n= 5^n\)
  5. \( \displaystyle a_n= 5 \cdot \left(\dfrac{2}{3}\right)^n\)
  6. \( \displaystyle a_n= \left( \dfrac{3}{2}\right)^n\)
  7. \( \displaystyle a_n=3^{-n}\)

3. Determine if the sequence converges or diverges. If it converges, determine its limit.

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  1. \(\displaystyle a_n= 4^{\frac{n}{2n+1}}\)
  2. \( \displaystyle a_n = \left(\dfrac{2n}{3n+1}\right)^2\)
  3. \( \displaystyle a_n= 3^{\frac{n}{n^2+1}}\)

4. Determine if the sequence converges or diverges. If it converges, determine its limit.

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  1. \( \left\{ 2, -\dfrac{4}{3}, \dfrac{8}{9},-\dfrac{16}{27}, \dfrac{32}{81}, \ldots \right\}\)
  2. \(\{-4, 12, -36, 108, - \ldots \} \)
  3. \(a_n=(-1)^n\)
  4. \( \displaystyle a_n=\dfrac{(-1)^{n+1}}{n}\)

5. Determine if the sequence converges or diverges. If it converges, determine its limit.

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  1. \( \displaystyle a_{n}=e^{-\frac{3}{\sqrt n}}\)
  2. \( \displaystyle a_{n}=\dfrac{(-1)^{n}}{5 \sqrt{n}}\)
  3. \( \displaystyle a_{n}=\dfrac{n^{2}}{\sqrt{n^{3}+7 n}} \)
  4. \( \displaystyle a_{n}= n^{2}\, e^{-n} \)
  5. \( \displaystyle a_{n}=\ln(3n^2+5)-\ln(n^2+1) \)

6. Determine if the sequence converges or diverges. If it converges, determine its limit.

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  1. \(\displaystyle a_n= \dfrac{5n^2+2n+20}{3n^5 -3n^2-5} \)
  2. \( \displaystyle a_n= \dfrac{3n^4 -2n}{8n^4+ 100n^2+1000}\)
  3. \( \displaystyle a_n=\dfrac{2n^3+1}{1500n^2+100n+3000}\)

7. Determine if the sequence converges or diverges. If it converges, determine its limit.

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  1. \( \displaystyle a_n= 2^{-n} \)
  2. \( \displaystyle a_n= 3^{n}\, 2^{1-n} \)
  3. \(\displaystyle a_n=4^{n} \)
  4. \( \displaystyle a_n= 2^{\frac{2n}{n+1}} \)