QUIZ 1 SOLUTIONS
(USING 9-10 DISCUSSION VERSION)

(note: i will use the symbol * for infinity)

a)       ln(x)                1/x
   lim   ----- = (by l'h) lim --- = 0
   x->*    x                   1


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b) you were given in class that

                               x     c
                  lim (1 + c/x)   = e
                  x->*

   so the answer is
                         2
                        e .
                                      x                    ln (1+2/x)
   alternatively, let y = ln [ (1+2/x) ] = x ln (1+2/x) =  ---------- .
                                                               1/x
   then 
                      ln (1+2/x)
         lim  y = lim ----------  which is a 0/0 form, so using l'hopital's
         x->*            1/x

   rule,
                     -2
                  -2x
                 -------
                 1 + 2/x               2
         = lim -----------   =  lim -------  =  2,
                     -2             1 + 2/x
                   -x

   so 
             y    2
        lim e  = e .
                     
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c) (excuse the bad attempt at sigma)

   _*_
   \    -n
   /__ 2    is a geometric series with common ratio 1/2 and initial term 1/2.
   n=1

   therefore, it equals 

                            1/2
			  -------  =  1
			  1 - 1/2

   by the geometric series formula.

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d) 

   _*__
   \        n                                           n
    \   (-1)                                         (-1)
    /   -----   is an alternating series. since lim  ----- = 0, the 
   /___ ln(n)                                   n->* ln(n)
    n=2

   series converges by the alternating series test.