• Homework ( Solutions )
  1. 
     

     ¥ å n = 1  4 (4n-3)(4n+1) .

  2. 1+3+¼+243. (have to be able to apply formula, easy enough!)
  3. 1.414141414...=as a fraction.

• Series
• Sum of sequence elements.
• Geometric sequence: S_n =a+ar++ar^n-1
  rS_n =ra++ar^n-1+ar^n
  (1-r)S_n =a-ar^n
  S_n =a(1-r^n)
• Telescoping series: 1 =1/k-1/(k+1)
  s_k =(1/1-1/2)+(1/2-1/3)+(1/k-1/(k+1))=1-1/(k+1). So infinite series goes to 1.
• diverges if | r | greater than or equal to one, converges otherwise.
• 3.333333=3(1+1/10+1/100+1/1000+¼) is a series with r < 1 so converges.
  

  ¥ å n = 1  arn-1 = a 1-r .

• nth-term test for divergence if have sequence a₀,...,a_n then
  

  an = sn-sn-1 ® 0,

  if the infinite sum is to converge. This means that 1-1+1-1+¼ converges since the limit for a_n does not exist.
• Properties of convergent series:sum rule, difference rule, constant multiple rule.

Next lecture

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File translated from T_EX by T_TH, version 2.70.
On 3 May 2002, 13:15.