Home
Class 11
MATHS
The sum to infinity of the series 1+2...

The sum to infinity of the series `1+2/3+6/(3^2)+(10)/(3^3)+(14)/(3^4). . . . . .` is (1) 2 (2) 3 (3) 4 (4) 6

A

2

B

3

C

4

D

6

Text Solution

Verified by Experts

The correct Answer is:
B
Let S be the sum of the given series i.e.,
`S=1+(2)/(3)+(6)/(3^(2))+(10)/(3^(3))+(14)/(3^(4))+ . . . . . "to "oo`
`rArr" "(S-1)=(2)/(3)+(6)/(3^(2))+(10)/(3^(3))+(14)/(3^(4))+ . . . . ."to "oo` . . .(i)
`rArr" "(S-1)xx(1)/(3)=(2)/(3^(2))+(6)/(3^(3))+(10)/(3^(4))+ . . . "to "oo` . . . .(ii)
Subtracting (ii) from (i) we get
`(2)/(3)(S-1)=(2)/(3)+(3^(4/2))/(1-(1)/(3))rArr(2)/(3)(S-1)=(2)/(3)(S-1)=(2)/(3)+(2)/(3)rArrS=3`
ALITER From (i), we observe that 3(S-1) is an infinite arithmetic- gemoetric series with a=2, d=4 and `r=(1)/(3)`.
`:." "3(S-1)=(1)/(1-(1)/(3))+(4xx(1)/(3))/((1-(1)/(3))^(2))=3+3rArrS=3`.
Promotional Banner

Similar Questions

Explore conceptually related problems

The sum to infinity of the following series (1)/(1*2)+(1)/(2*3)+(1)/(3*4)+.... shall be

The sum of the series (1)/(3)+(1)/(3^(2)xx2)+(1)/(3^(3)xx3)+(1)/(3^(4)xx4)+... is :

If the sum to infinity of the series 3 + (3 + d). (1)/(4) + (3 + 2d) (1)/(4^(2)) + ..... " is " (44)/(9) then find d

If the sum to infinity of the series 3+(3+d)(1)/(4)+(3+2d)(1)/(4^(2))+...oo is (44)/(9) then find d.

The sum of the series 1+ 2/3+ (1)/(3 ^(2)) + (2 )/(3 ^(3)) + (1)/(3 ^(4)) + (2)/(3 ^(5)) + (1)/(3 ^(6))+ (2)/(3 ^(7))+ …… upto infinite terms is equal to :