Math 381                                                                                            Name:  KEY
Quiz 7 -- Take Home due Friday, November 8 at 11:45. You may use any theorem in the book as long as you reference it.
  1. Use the fact that  MATH to find the exact value of each of the following:
    1. MATH

      2. MATH
       
    2. MATH

      3. MATH
       
  2.         a.  Let MATH be a sequence of real numbers with $t_{n}>0$ for all $n\in \QTR{Bbb}{N}.$ Prove that if $\lim t_{n}=0$ then MATH
      3. MATH
       
    1. Prove: If $a_{n}>0$ for all $n\in \QTR{Bbb}{N}$ and $\sum a_{n}$ converges then MATH diverges.

      2. MATH
  3. Prove: If MATH for all $n\in \QTR{Bbb}{N}$ then $\sum a_{k}$ diverges.

    4. MATH
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