Home
Class 12
MATHS
If S(n)=1+1/2+1/3+…+1/n(ninN), then prov...

If `S_(n)=1+1/2+1/3+…+1/n(ninN)`, then prove that
`S_(1)+S_(2)+..+S_((n-1))=(nS((n))-n)or(nS((n-1))-n+1)`

Text Solution

Verified by Experts

`S_((1))+S_((2))+..+S_((n-1))`
1+
`1+1/2+`
`1+1/2+1/3`
………………..
………………..
`1+1/2+1/3+………1/(n-1)`
Adding vertically
`=(n-1)+((n-2))/2+((n-3))/3+….((n-(n-1))/((n-1)))`
`=n[1+1/2+1/3+…….+1/(n-1)]-[1+1+1+….1]`
`=nS_((n-1))-(n-1)=nS_(n)-n`
Promotional Banner

Topper's Solved these Questions

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLES 5.7|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLES 5.8|1 Videos

  • PROGRESSION AND SERIES

    CENGAGE PUBLICATION|Exercise SOLVED EXAMPLES 5.5|1 Videos

  • PROBABILITY II

    CENGAGE PUBLICATION|Exercise MULTIPLE CORRECT ANSWER TYPE|6 Videos

  • PROPERTIES AND SOLUTIONS OF TRIANGLE

    CENGAGE PUBLICATION|Exercise Archives (Numerical Value Type)|3 Videos

Similar Questions

Explore conceptually related problems

For ninN , prove that ((n+1)/2)^ngt n!

For n in N , Prove that (n+1)[n!n+(n-1)!(2n-1)+(n-2)!(n-1)]=(n+2)!

If S_(1), S_(2), S_(3) be respectively the sums of n, 2n and 3n terms of a G.P., prove that, S_(1)(S_(3) - S_(2)) = (S_(2) - S_(1))^(2) .

If ninN,xgt-1 and (1+x)^(n)ge1+nx , then prove that (1+x)^(n+1)ge1+(n+1)x .

If S_(1), S_(2), S_(3),…., S_(n) are the sums to infinity of n infinte geometric series whose first terms are 1,2,3,… n and whose common ratios are (1)/(2), (1)/(3), (1)/(4), ….(1)/(n+1) respectively, show that, S_(1) + S_(2) + S_(3) +…S_(n) = (1)/(2)n(n+3)

If a_1. a_2 ....... a_n are positive and (n - 1) s = a_1 + a_2 +.....+a_n then prove that (a_1 + a_2 +....+a_n)^n ge (n^2 - n)^n (s - a_1) (s - a_2)........(s - a_n)

Prove that n^n > 1, 3, 5,…………(2n - 1) .

If S_(n) be the sum of n consecutive terms of an A.P. show that, S_(n+3) - 3S_(n+2) + 3S_(n+1) - S_(n) = 0

For all n ge 1 prove that (1)/(1.2)+ (1)/(2.3)+(1)/(3.4)+.....+(1)/(n(n+1))=(n)/(n+1)

If S=a_1+a_2+......+a_n,a_i in R^+ for i=1 to n, then prove that S/(S-a_1)+S/(S-a_2)+......+S/(S-a_n) ge n^2/(n-1), AA n ge 2