Home
Class 12
MATHS
Prove that sum(r=1)^n(-1)^(r-1)(1+1/2+1...

Prove that `sum_(r=1)^n(-1)^(r-1)(1+1/2+1/3++1/r)(.^n C_r)=1/n` .

Text Solution

Verified by Experts

`sum(-1)^(r-1).^(n)C_(r)(1/1+1/2+1/3+"...."+1/r)`
`=sum((-1)^(r-1).^(n)C_(r)underset(0)overset(1)int(1+x+x^(2)+"....."+x^(r-1))dx)`
`=sum(-1)^(r-1)..^(n)C_(r)underset(0)overset(1)int((1-x^(r))/(1-x))dx`
`= underset(0)overset(1)intunderset(r=1)overset(n)sum((-1)^(r-1)..^(n)C_(r)-(-1)^(r-1)..^(n)C_(r)x^(r))/(1-x)dx`
`=underset(0)overset(1) int(.^(n)C_(0)+(1-x)^(n) + (1-x)^(n))/(1-x)dx`
`= underset(0)overset(1)int(1-x)^(n-1)dx`
`= [(-(1-x)^(n-1))/(n-1)]_(0)^(1)`
`= 1/n`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Example|10 Videos

  • BINOMIAL THEOREM

    CENGAGE PUBLICATION|Exercise Concept Application Exercise 8.1|17 Videos

  • AREA

    CENGAGE PUBLICATION|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE PUBLICATION|Exercise For problems 3 and 4|2 Videos

Similar Questions

Explore conceptually related problems

Prove that sum_(r=1)^(n) tan^(-1) ((2^(r -1))/(1 + 2^(2r -1))) = tan^(-1) (2^(n)) - (pi)/(4)

Prove that sum_(r=1)^(k) (-3)^(r-1) (3n)^C_(2r-1) = 0 , where k = 3n//2 and n is an even integer.

prove that sum_(r=0)^n(-1)^r^n C_r . [ 1/(2^r)+(3^r)/(2^(2r))+(7^r)/(2^(3r))+(15^r)/(2^(4r))+ ......up to m terms ] = (2^(m n)-1)/(2^(m n)(2^n-1))

The value of sum_(r=1)^(n)(-1)^(r-1)((r )/(r+1))*^(n)C_(r ) is (a) 1/(n+1) (b) 1/n (c) 1/(n-1) (d) 0

Prove that (3!)/(2(n+3))=sum_(r=0)^n(-1)^r((n C_r)/((r+3)C_3))

Prove that sum_(r=0)^n^n C_r(-1)^r[i+i^(2r)+i^(3r)+i^(4r)]=2^n+2^(n/2+1)cos(npi//4),w h e r ei=sqrt(-1)dot

Prove that .^nP_r = ^(n-1)P_r + r^(n-1)P_(r-1)

Prove that sum_(r=0)^ssum_(s=1)^n^n C_s^s C_r=3^n-1.

Prove that .^(n)P_(r)=.^(n-1)P_(r)+r.^(n-1)P_(r-1) .

Find the sum sum_(r=1)^n r(r+1)(r+2)(r+3)