Home
Class 12
MATHS
Prove that (C1)/1-(C2)/2+(C3)/3-(C4)/4++...

Prove that `(C_1)/1-(C_2)/2+(C_3)/3-(C_4)/4++((-1)^(n-1))/n C_n=1+1/2+1/3++1/ndot`

Text Solution

Verified by Experts

We have,
`(1+x)^(n)=.^(n)C_(0)+.^(n)C_(1)x+.^(n)C_(2)x^(2)+"....."+.^(n)C_(n)x^(n)`
Dividing by x, we get
`((1-x)^(n)-1)/(x) = .^(n)C_(1) + .^(n)C_(2)x+"..... "+.^(n)C_(n)x^(n-1)`
`:. 1+(1+x)+(1+x)^(2) + "...."+(1+x)^(n-1)`
`= .^(n)C_(1) + .^(n)C_(2)x + "..." + .^(n)C_(n)x^(n-1)`.
`:. underset(-1)overset(0)int(.^(n)C_(1) + .^(n)C_(2)x+C_(3)x^(2) + "...."+.^(n)C_(n) x^(n-1))dx`
`= underset(-1)overset(0)int[1+(1+x)+"....."+(1+x)^(n-1)]dx`
`rArr [.^(n)C_(1)x+(.^(n)C_(2)x^(2))/(2)+(.^(n)C_(3)x^(2))/(3)+"...."+(.^(n)C_(n)x^(n))/(n)]_(-1)^(0)`
`= [x+((1+x)^(2))/(2)+((1+x)^(3))/(3)+"....."+((1+x)^(n))/(n)]_(-1)^(0)`
`rArr (.^(n)C_(1))/(1)-(.^(n)C_(2))/(2)+(.^(n)C_(3))/(3)-"...."+((-1)^(n-1))/(n).^(n)C_(n)=1+1/2+1/3+"......"+1/n`
Promotional Banner

Topper's Solved these Questions

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Example|10 Videos

  • BINOMIAL THEOREM

    CENGAGE ENGLISH|Exercise Concept Application Exercise 8.1|17 Videos

  • AREA

    CENGAGE ENGLISH|Exercise Comprehension Type|2 Videos

  • CIRCLE

    CENGAGE ENGLISH|Exercise MATRIX MATCH TYPE|7 Videos

Similar Questions

Explore conceptually related problems

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + …+ C_(n) x^(n) , show that C_(1) - (C_(2))/(2) + (C_(3))/(3) - …(-1)^(n-1) (C_(n))/(n) = 1 + (1)/(2) + (1)/(3) + …+ (1)/(n) .

If (1 + x)^(n) = C_(0) + C_(1) x + C_(2) x^(2) + C_(3) x^(3) + … + C_(n) x^(n) , prove that C_(0) - (C_(1))/(2) + (C_(2))/(3) -…+ (-1)^(n) (C_(n))/(n+1) = (1)/(n+1) .

Prove that : C_(0)-3C_(1)+5C_(2)- ………..(-1)^n(2n+1)C_(n)=0

Show that (C_(0))/(1) - (C_(1))/(4) + (C_(2))/(7) - … + (-1)^(n) (C_(n))/(3n +1) = (3^(n) * n!)/(1*4*7…(3n+1)) , where C_(r) stands for ""^(n)C_(r) .

Prove that (.^n C_0)/1+(.^n C_2)/3+(.^n C_4)/5+(.^n C_6)/7+ . . . =(2^n)/(n+1)dot

Prove that nC_1(nC_2)^2(n C_3)^3.......(n C_n)^n le ((2^n)/(n+1))^((n+1)C_2),AAn in Ndot

Prove that C_(1)^(2)-2*C_(2)^(2)+3*C_(3)^(2)-…-2n*C_(2n)^(2)=(-1)^(n)n*C_(n)

Prove that 1/(n+1)=(.^n C_1)/2-(2(.^n C_2))/3+(3(.^n C_3))/4- . . . +(-1^(n+1))(n*(.^n C_n))/(n+1) .

Prove that C_1/C_0+(2c_(2))/C_1+(3C_3)/(C_2)+......+(n.C_n)/(C_(n-1))=(n(n+1))/2

Prove that (.^(n)C_(1))/(2) + (.^(n)C_(3))/(4) + (.^(n)C_(5))/(6) + "…." = (2^(n) - 1)/(n+1) .