________________________________________________________________________________
Calculus  II                                                      Quiz #9                                                       May 6, 2003

Name____________________                  R.  Hammack                                                Score ______
_______________________________________________________________________________

Decide if the following series converge or diverge. In the case of convergence, state the sum, if possible.


(1)   2/5^0 + 2/5^1 + 2/5^2 + 2/5^3 + 2/5^4 + ... + 2/5^k +...

This is a geometric series with a = 2 and r = 1/5.
Since |r| < 1, it converges to a/(1 - r) = 2/(1 - 1/5) = 2/(4/5) = 5/2



(2)    2/1^5 + 2/2^5 + 2/3^5 + 2/4^5 + ... + 2/k^5 +...

This is Underoverscript[∑ , k = 2, arg3] 2/k^5 = 2Underoverscript[∑ , k = 2, arg3] 1/k^5, which is twice a p-series with p = 5 >1

Therefore it converges.



(3)    Underoverscript[∑ , k = 2, arg3] (k^2 + k + 3)/(2k^2 + 1)

Note Underscript[lim , k ∞] (k^2 + k + 3)/(2k^2 + 1) = 1/2≠0, so the series diverges, by the divergence test.


(4)     Underoverscript[∑ , k = 0, arg3] (1/2^k + 2/3^k)

Underoverscript[∑ , k = 0, arg3] (1/2^k + 2/3^k) = Underoverscript[∑ , k = 0, arg3 ... + Underoverscript[∑ , k = 0, arg3] 2 (1/3)^k = 1/(1 - 1/2) + 2/(1 - 1/3) = 1/(1/2) + 2/(2/3)
  
  = 2 + 3 = 5