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Calculus  II                                           Quiz #7                     April 29, 2005

Name_________________            R.  Hammack                  Score ______
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Decide if the following series converge or diverge. In the case of convergence state the sum, if possible.

(1)   Underoverscript[∑ , k = 1, arg3](3/k-3/(k + 1)) = Underscript[lim , n∞]s_n = Underscript[lim , n∞](3/1-3/(n + 1)) = 3-0 = 3

Note, this is a telescoping series, since:
s_n= (3/1-3/2) + (3/2-3/3) + (3/3-3/4) + (3/4-3/5) +...+ (3/n-3/(n + 1)) = 3/1-3/(n + 1)


(2)   (5/7)^2+(5/7)^3+(5/7)^4+(5/7)^5+(5/7)^6+(5/7)^7+ ...

Geometric series with a =25/49and r =5/7< 1.

CONVERGES to 25/49/(1 - 5/7)=25/49/2/7=25/497/2=25/14

(3)    5/7+5/8+5/9+5/10+5/11+5/12+ ... =   5(1/7+1/8+1/9+1/10+1/11+1/12+ ...)

Harmonic series with first 6 terms deleted.  DIVERGES


(4)    5/7^4+5/8^4+5/9^4+5/10^4+5/11^4+5/12^4+ ...

P-series with p = 4 > 1.  CONVERGES



(5)    5/7^(1/2)+5/8^(1/2)+5/9^(1/2)+5/10^(1/2)+5/11^(1/2)+5/12^(1/2)+ ...

P-series with p = 1/2 < 1.  DIVERGES