For example if k 1 then the series at that value

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values to be negative. For example, if  k= -1, then the series at that value would be 1 3 (− 1 )+ 1 = 1 2 .  In order for  k  to be positive,  k  will have to be greater than values that  would cause the series to be negative or discontinuous. Since  k =1, the denominator will be  positive, thus making the series positive. Lastly, the series will need to be evaluated to 
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determine if it is a decreasing function. Since  1 3 k + 1  is differentiable, we can evaluate the  derivative to determine if the series is decreasing for  f’(k) 1) f ( k ) = 1 3 k + 1  or  ( 3 k + 1 ) 1      d dk ( 3 k + 1 ) 1 2) Chain Rule:  −( 3 k + 1 ) 2 3 3) Rewrite:  f ’ ( k ) = 3 ( 3 k + 1 ) 2    Since the numerator is negative, the denominator is raised to an even power and non- negative, we have a negative divided by a positive giving us a negative value, and  f’(k)  is  always negative except when  k=  - 1 3 , which would make the series discontinuous at that  value. Thus, the series will always be decreasing.  C. Given:  k = 1 1 3 k + 1 Integral Test: 1) 1
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  • Summer '17
  • lim, Indian mathematics

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