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Fine the limit of (x^1000 - 1)/(x - 1) as x approaches 1.
Solution:
(x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)
= (x^999 + x^998 + ... + x + 1)
Substituting x = 1 I get....
Line (1):1 + 11 + 12 + ..... + 1999 = 1 + 999 = 1000
Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and with out obviously counting it on my fingers.
Solution:
(x^1000 - 1)/(x - 1) = (x^999 + x^998 + ... + x + 1)(x - 1)/(x - 1)
= (x^999 + x^998 + ... + x + 1)
Substituting x = 1 I get....
Line (1):1 + 11 + 12 + ..... + 1999 = 1 + 999 = 1000
Is there any way to prove Line (1): I know it is obvious but I want to prove it mathematically and with out obviously counting it on my fingers.