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Question
Evaluate the following :
`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`
Solution
`lim_(x -> 1) [(x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n" - "n"^2)/(x - 1)]`
1 + 3 + 5 + … + (2n – 1)
= `sum_("r" = 1)^"n" (2"r" - 1)`
= `2 sum_("r" = 1)^"n" "r" - sum_("r" = 1)^"n" 1`
= `2("n"("n" + 1))/2 - "n"`
= n(n + 1) – n
= n2 + n – n
= n2
∴ n2 = 1 + 3 + 5 + … + (2n – 1).
∴ Required limit
= `lim_(x -> 1) ([ x + 3x^2 + 5x^3 + ... + (2"n" - 1)x^"n"] - [1 + 3 + 5 + ... + (2"n" - 1)])/(x - 1)`
= `lim_(x -> 1) ((x - 1) + (3x^2 - 3) + (5x^2 - 5) + ... + (2"n" - 1)x^"n" - (2"n" - 1))/(x - 1)`
= `lim_(x -> 1) [(x - 1)/(x - 1) + (3(x^2 - 1))/(x - 1) + (5(x^3 - 1))/(x - 1) + ... + ((2"n" - 1)(x^"n" - 1))/(x - 1)]`
= `lim_(x -> 1) ((x^1 - 1^1)/(x - 1)) + 3lim_(x -> 1) ((x^2 - 1^2)/(x - 1)) + 5lim_(x -> 1)((x^3 - 1^3)/(x - 1)) + ... + (2"n" - 1) lim_(x -> 1) ((x^"n" - 1^"n")/(x - 1))`
= 1(1)0 + 3(2)(1)1 + 5(3)(1)2 + … + (2n – 1) n(1)n–1
= 1(1) + 3(2) + 5(3) + … + (2n – 1)n
= `sum_("r" = 1)^"n" (2"r" - 1)"r"`
= `sum_("r" = 1)^"n" (2"r"^2 - "r")`
= `2 sum_("r" = 1)^"n" "r"^2 - sum_("r" = 1)^"n" "r"`
= `2* ("n"("n" + 1)(2"n" + 1))/6 - ("n"("n" + 1))/2`
= `"n"("n" + 1)((2"n" + 1)/3 - 1/2)`
= `"n"("n" + 1) ((4"n" + 2- 3)/6)`
= `("n"("n" + 1) (4"n" - 1))/6`
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