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Let `S_k ,k=1,2, ,100 ,`denotes thesum of the infinite geometric series whose first term s `(k-1)/(k !)`and the common ratio is `1/k`, then the value of `(100^2)/(100 !)+sum_(k=2)^(100)(k^2-3k+1)S_k`is _______. |
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Answer» `S_(k)=(a)/(1-r)=((k-1)/(k!))/(1-(1)/(k))=(k)/(k!)=(1)/((k-1)!)` Now, `sum_(k=2)^(100)|(k^(2)-3k+1)S_(k)|=sum_(k=2)^(100)|(k^(2)-3k+1)*(1)/((k-1)!)|` `=sum_(k=2)^(100)|((k-1))/((k-2))=(k)/((k-1)!)|` `=|(1)/(0!)-(2)/(1!)|+|(2)/(1!)-(3)/(2!)|+|(3)/(2!)-(4)/(3!)|+"........."+|(99)/(98!)-(100)/(99!)|` `=((2)/(1!)-(1)/(0!))+((2)/(1!)-(3)/(2!))+((3)/(2!)-(4)/(3!))+"........."+((99)/(98!)-(100)/(99!))` `=3-(100)/(99)=3-((100)^(2))/(100!)` `:.((100)^(2))/(100!)+sum_(k=2)^(100)|(k^(2)-3k+1)S_(k)|=3`. |
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