Home
Class 12
MATHS
Evaluate the limit: ("lim")(nvecoo)((1^2...

Evaluate the limit: `("lim")_(nvecoo)((1^2-2^2+3^3-4^2+5^2+ nt e r m s)/(n^2)`

Text Solution

Verified by Experts

The correct Answer is:
`1//2" or "-1//2`
When n is even:
Given series
`1^(2)-2^(2)+3^(2)-4^(2)+...-n^(2)`
`(1^(2)-2^(2))+(3^(2)-4^(2))+...[(n-1)^(2)-n^(2)]`
`=-(1+2+3+4+...+n)`
`=-(n(n+1))/(2)`
`:. "Given "L=underset(ntooo)lim(n(n+1))/(2n^(2))=-1/2`
When n is odd:
Given series
`1^(2)-2^(2)+3^(2)-4^(2)+...+n^(2)`
`=-1(1+2+3+...+(n-1))+n^(2)`
`=-(n(n-1))/(2)+n^(2)`
`=(n(n+1))/(2)`
`:." Given "L=underset(ntooo)lim(n(n+1))/(2n^(2))=1/2`
Promotional Banner

Topper's Solved these Questions

  • LIMITS

    CENGAGE PUBLICATION|Exercise EXERCISE 2.4|5 Videos

  • LIMITS

    CENGAGE PUBLICATION|Exercise EXERCISE 2.5|12 Videos

  • LIMITS

    CENGAGE PUBLICATION|Exercise EXERCISE 2.2|7 Videos

  • JEE 2019

    CENGAGE PUBLICATION|Exercise Chapter 10|8 Videos

  • LINEAR COMBINATION OF VECTORS, DEPENDENT AND INDEPENDENT VECTORS

    CENGAGE PUBLICATION|Exercise DPP 1.2|10 Videos

Similar Questions

Explore conceptually related problems

Evaluate the limit: ("lim")_(nrarroo)((1^2-2^2+3^2-4^2+5^2+ nt e r m s))/(n^2)

Evaluate the limit: ("lim")_(x->oo)[sqrt(a^2x^2+a x+1)-sqrt(a^2x^2+1)]

Evaluate the limit: lim_(x->1)((2x-3)(sqrt(x)-1))/(2x^2)

Evaluate the limit lim_(x->0) (e^(2x)+e^(-x)-2)/x

Evaluate the following limits : lim_(nrarrinfty)(1^2+2^2+3^2+...+n^2)/(2n^3)

Find the value of the limit lim_(nrarroo)(1^2+2^2+...+n^2)/n^3

Evaluate: ("lim")_(n to oo)((1^2+2^2+3^3++n^2)(1^3+2^3+3^3++n^3))/(1^6+2^6+3^6++n^6)

Evaluate : lim_(xrarroo)(1+2+3+...+n)/n^2

Evaluate the limit: underset(x to0)lim(e^(2x)+e^(-x)-2)/(x)

Evaluate the following limits : lim_(xrarr2) (x^(7/3)-2^(7/3))/(x-2)