Question

limit tends to 1 (x+x^2+x^3+.....+x^n-n)/x-1

Answer 1

Lim(x→1) [x + x² + x³ +.........+ xⁿ - n]/(x - 1)

First of all we have to check form of limit ,put x = 1 . we get 0/0 is the form of limit here.

here, we use L-HOSPITAL Rule ,
differentiate separately numerator and denominator with respect to x

Lim(x→1) [ 1 + 2x + 3x² + ......nx^(n-1) -0]/(1 -0)
now, put the value of x = 1
we get ,
1 + 2 + 3 + ...... + n = n(n + 1)/2 [ we know, sum of natural numbers of n terms = n(n + 1)/2]

hence, answer is n(n + 1)/2

Answer 2

Answer:

Step-by-step explanation:

The above problem, if looked at a bit more carefully gives this:

limx→1(x−1)+(x2−1)+(x3−1)+…+(xn−1)x−1

This simplified, gives us:

limx→11+(x+1)+(x2+x+1)+…+(xn−1+xn−2+…+x+1)=1+2+3+…+n=n(n+1)2

We could go by the Gemoetric Progression formula too:

limx→1x(xn−1)x−1−nx−1=limx→1xn+1−(n+1)x+n(x−1)2

Now we can use the L'Hopital's rule twice (after inspection) and proceed, we get the same answer as above.