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Question
Evaluate the following limit :
`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`
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Solution
`lim_(x -> 1)[(x + x^2 + x^3 + ......... + x^"n" - "n")/(x - 1)]`
= `lim_(x -> 1)[(x + x^2 + x^3 + .... + x^"n" - (1 + 1 + 1 + ... "n times"))/(x - 1)]`
= `lim_(x -> 1) ((x - 1) + (x^2 - 1) + (x^3 - 1) + ... + (x^"n" - 1))/(x - 1)`
= `lim_(x -> 1)[(x^1 - 1^1)/(x - 1) + (x^2 - 1^2)/(x - 1) + (x^3 - 1^3)/(x - 1) + ... + (x^"n" - 1^"n")/(x - 1)]`
= 1 (1)0 + 2 (1)1 + 3 (1)2 + 4 (1)3 + ... + n (1)n–1 ...`[lim_(x -> "a") (x^"n" - "a"^"n")/(x - "a") = "na"^("n" - 1)]`
= 1 + 2 + 3 + 4 + ... + n
= `("n"("n" + 1))/2`
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