Home
Class 12
MATHS
IF f(r )=1+(1)/(2)+(1)/(3)+"...."+(1)/(r...

IF `f(r )=1+(1)/(2)+(1)/(3)+"...."+(1)/(r )` and `f(0)=0`, find `sum _(r=1)^(n)(2r+1)f(r )`.

Text Solution

Verified by Experts

Since, `sum _(r=1)^(n)(2r+1)f(r )`
`=sum _(r=1)^(n)(r^(2)+2r+1-r^2))f(r )=sum _(r=1)^(n){(r+1)^(2)-r^(2)}f(r )`
`=sum _(r=1)^(n){(r+1)^(2)f(r )-(r+1)^(2)f(r+1)+(r+1)^(2)f(r+1)-r^(2)f(r )}`
`=sum _(r=1)^(n)(r+1)^(2){f(r )-(r+1)}+sum _(r=1)^(n){(r+1)^(2)f(r+1)-r^(2)f(r )}`
`=-sum _(r=1)^(n)((r+1)^(2))/((r+1))+sum _(r=1)^(n)(r+1)^(2)f(r+1)+(n+1)^(2) f(n+1)-sum_r^(r=1)r^(2)f(r)[:.f(r+1)-f(r)=(1)/(r+1)]`
`=-sum _(r=1)^(n)(r+1)+{2^(2)f(2)+3^(2)f(3)+"...."n^(2)f(n)}+(n+1)^(2)f(n+1)={1^(2)f(1)+2^(2)f(2)+3^(2)+"...."n^(2)f(n)}`
`=-sum _(r=1)^(n)r-sum _(r=1)^(n)1+(n+1)^(2)f(n+1)-1^(2)f(1)`
`=(n(n+1))/(2)-n+(n+1)^(2)f(n+1)-f(1)`
`=(n+1)^(2)f(n+1)-(n(n+1))/(2)-1 " "[:.f(1)=1]`
`=(n+1)^(2)f(n+1)-((n^(2)+3n+2))/(2)`
Hence, this is the required result.
Promotional Banner

Similar Questions

Explore conceptually related problems

The value of sum_(r=1)^(n)(nP_(r))/(r!)=

If t_(1)=1,t_(r )-t_( r-1)=2^(r-1),r ge 2 , find sum_(r=1)^(n)t_(r ) .

If sum_(r=1)^(n) t_(r ) = sum_(k=1)^(n) sum_(j=1)^(k) sum_(i=1)^(j) 2 , then sum_(r=1)^(n) (1)/( t_(r )) equals :

Sum of the series sum_(r=1)^(n) (r^(2)+1)r! is

If a_(1),a_(2),".....", are in HP and f_(k)=sum_(r=1)^(n)a_(r)-a_(k) , then 2^(alpha_(1)),2^(alpha_(2)),2^(alpha_(3))2^(alpha_(4)),"....." are in {" where " alpha_(1)=(a_(1))/(f_(1)),alpha_(2)=(a_(2))/(f_(2)),alpha_(3)=(a_(3))/(f_(3)),"....."} .

lim_(n rarr oo) sum_(r=0)^(n-1) 1/(n+r) =

If sum_(r = 1)^(k) cos^(-1) beta r = (kpi)/(2), "for any k" ge 1 and A = sum_(r = 1)^(k)(beta r)^(r),"then" lim_(x rarr A) ((1 + x^(2))^(1//3)-(1-2x)^(1//4))/(x+x^(2)) is equal to :

For r=0,1, . . . ,10 let A_(r),B_(r) and C_(r) denote respectively, the coefficient of x^(r) in the expansions of : (1+x)^(10),(1+x)^(20) and (1+x)^(30) . Then sum_(r=1)^(10)(B_(10)B_(r)-C_(10)A_(r)) is equal to:

If f:RtoR satisfies f(x+y)=f(x)+f(y) for all x,yinRandf(1)=7 , then sum_(r=1)^(n)f(r) is :

lim_(n rarr oo) (1)/(n^(3)) sum_(r = 1)^(n) r^(2) is :