1. See definitions in book.

    2.  
  2. See statements in book.

    3.  
  3. For each description, either give an example that fits the description or explain why it is impossible to do so.

    4.  
    1. A convergent series $\sum a_{n}$ with $a_{n}>0$ such that MATH is divergent.

      2. MATH
       
    2. A divergent series $\sum a_{n}$ with $a_{n}>0$ such that MATH is convergent.

      3. MATH
       
    3. A function with domain equal to $\QTR{Bbb}{R}$ that is discontinuous at $1$ and $3.$

      4. MATH
       
  4. Determine if each of the following are convergent or divergent. $\ $You may use any of the Series Tests from the book, but make sure to demonstrate your reasoning.

    5.  
    1. MATH

      2. MATH
       
    2. MATH

      3. MATH
       
  5. Prove directly from the definition of continuity that MATH is discontinuous at $x=3.$

    6. MATH
     
     
  6. Prove using the MATH property of continuity that $f(x)=x^{3}$ is continuous at $x=0.$

    7. MATH
     
  7. Calculate MATH if:
    1. MATH and $a_{n}=\dfrac{1}{n}$ for $n\leq 10.$

      2. MATH
       
    2. $\lim a_{n}=7.$

      3. MATH
       
    3. The partial sums MATH are given by the formula MATH

      4. MATH
       
  8. Let $f$ be a continuous function such that dom $f=\QTR{Bbb}{R}$ and $f(r)=0$ for all $r\in \QTR{Bbb}{Q}.$$\ $Prove that MATH

    9. MATH

    MATH

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