Home
Class 12
MATHS
sum(i=1)^(oo)sum(j=1)^(oo)sum(k=1)^(oo)(...

`sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i+j+k))` is equal to (where `|a| gt 1`)

A

`(a-1)^(-3)`

B

`(3)/(a-1)`

C

`(3)/(a^(3)-1)`

D

None of these

Text Solution

Verified by Experts

The correct Answer is:
A
`(a)` `S=sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i+j+k))`, `|a| gt 1` or `0 lt (1)/(|a|) lt 1`
`=sum_(i=1)^(oo)sum_(j=1)^(oo)sum_(k=1)^(oo)(1)/(a^(i)a^(j)a^(k))`
`=(sum_(i=1)^(oo)(1)/(a^(i)))(sum_(j=1)^(oo)(1)/(a^(j)))(sum_(k=1)^(oo)(1)/(a^(k)))`
`=((1)/(a))/(1-(1)/(a))*((1)/(a))/(1-(1)/(a))*((1)/(a))/(1-(1)/(a))=(1)/((a-1)^(3))`
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE|Exercise Comprehension|7 Videos

  • PROGRESSION AND SERIES

    CENGAGE|Exercise Multiple Correct Answer|4 Videos

  • PROBABILITY II

    CENGAGE|Exercise JEE Advanced Previous Year|25 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE|Exercise JEE Advanced Previous Year|11 Videos

Similar Questions

Explore conceptually related problems

The sum sum_(k=1)^(20) k (1)/(2^(k)) is equal to

Find sum_(k=1)^(n)(1)/(k(k+1)) .

Find the sum_(k=1)^(oo) sum_(n=1)^(oo)k/(2^(n+k)) .

Find the value of underset((inejnek))(sum_(i=0)^(oo)sum_(j=0)^(oo)sum_(k=0)^(oo))1/(3^(i)3^(j)3^(k)) .

Find sum_(n=1)^(oo)1/(n^2+5n+6)

The number of real solution of the equation sin^(-1) (sum_(i=1)^(oo) x^(i +1) -x sum_(i=1)^(oo) ((x)/(2))^(i)) = (pi)/(2) - cos^(-1) (sum_(i=1)^(oo) (-(x)/(2))^(i) - sum_(i=1)^(oo) (-x)^(i)) lying in the interval (-(1)/(2), (1)/(2)) is ______. (Here, the inverse trigonometric function sin^(-1) x and cos^(-1) x assume values in [-(pi)/(2), (pi)/(2)] and [0, pi] respectively)

Find the sum sum_(j=1)^(10)sum_(i=1)^(10)ixx2^(j)