Need help with LIMITS?

Question

Hello guys

err i know i have to do my homework on my own but really believe me i have done trying this thing heap of times but i cant find do with it
pls help me ,,,,, i have find answer of

lim x-> 1

[(x) + (x)^2 + (x)^3 + ......... + (x)^n - n ] / (x - 1 )


i know the answer is n*(n+1) / 2 but how do you get it ???
can some one help me with evrything step by step ????


thanx a lottt in advance

Answer 1

Use L'Hopital's rule: if a function is equal to 0 or infinity at the limit, then the limit can be found by taking the derivative of the numerator and denominator.

So, take the derivative of your function:
f(x) = [(x) + (x)^2 + (x)^3 + ......... + (x)^n - n ] / (x - 1 )
f'(x) = [1 + 2x + 3x^2 + ....nx^(n-1)] / 1
f'(1) = [1 + 2 + 3 + ....n] (solution)

You can rewrite the answer as n * (n-1) / 2 (since 1 + 2 + 3 + .... + n = n * (n-1) / 2).