Question

((x+x^2+x^3........x^n)-n)/x-1​

Answer 1

Answer:

lim

x−1

x

1

+x

2

+x

3

+.....+x

n

−n

=

x→1

lim

x−1

(x

1

−1)+(x

2

−1)+(x

3

−1)+........+(x

n

−1)

=

x→1

lim

x−1

(x

1

−1)(1+(x+1)+(x

2

+x+1)+.........+(x

n

+x

n−1

+x

n−2

+........+1)

=

x→1

lim

(1+(x+1)+(x

2

+x+1)+.......+(x

n

+x

n−1

+x

n−2

+.....+1)

Applying limit x→1 we get ,

=1+2+3+ .......................... +n

=∑n

=

2

n(n+1)