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Algebra

Blazing goIITian

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6 Apr 2008 19:20:37 IST

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997

sequence and series doubt -->;

None

If f(r) = rac{1}{1} + rac{1}{2} + rac{1}{3} + .... + rac{1}{r}

and f(0) = 0

then find

_{[ r =1 ]}^{[ n ]} (2r + 1)f(r)

(please give complete solution)

EDIT -->

Ans -- > (n+1)² * f(n+1) - (n² + 3n + 2) / 2

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Comments (18)

arpan1

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6 Apr 2008 19:32:32 IST

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is the ans 2

anchit saini

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6 Apr 2008 19:35:38 IST

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huh?
how can u get an answer without getting "n" in it ???

arpan1

Blazing goIITian

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6 Apr 2008 19:38:43 IST

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i assumed n = infinity

anchit saini

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6 Apr 2008 19:40:55 IST

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what a great assumption !!!!

please solve it using n as a finite quantity

arpan1

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6 Apr 2008 19:47:33 IST

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see why dont u do it urself using limit of sum method !!!

anchit saini

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6 Apr 2008 19:48:40 IST

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arey do it naa yaar

i am not as intelligent as you are
thats why i am having a problem !!

Hari Shankar

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6 Apr 2008 19:52:53 IST

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Do u get (n+1)²

1/r - n(n+1)/2

anchit saini

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6 Apr 2008 19:55:29 IST

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sir the ans given is

(n+1)^2 * f(n+1) - (n^2 + 3n + 2) / 2

arpan1

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6 Apr 2008 19:56:53 IST

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_{[r=1 ]}

^{[n ]} 2r . f(r) + _{[r=1 ]}

^{[n ]} f(r)

=( n (n+1) + 1 ) _{[r=1 ]}

^{[n ]} f(r)

now as far as i know , f(r) forms an h.p. whose sum cannot be evaluated

anchit saini

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6 Apr 2008 20:09:46 IST

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arpan try to match your answer with the answer i gave

Hari Shankar

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6 Apr 2008 20:10:34 IST

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@anchit: the two answers are equivalent

(n+1)²f(n+1) - (n+1)(n+2)/2 = (n+1)²f(n) + (n+1) - (n+1)(n+2)/2 = (n+1)²f(n) - n(n+1)/2

arpan1

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6 Apr 2008 20:11:35 IST

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@ ACHIT

if u want, i will do it

anchit saini

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6 Apr 2008 20:16:27 IST

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ok sir

i didn't try matching it

pls give the soln

Hari Shankar

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6 Apr 2008 20:25:32 IST

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We are required to find $sum_{r=1}^n (2r+1) sum_{k=1}^r 1/k$

Just change the order of summation

You get $sum_{k=1}^n rac {1} {k} sum_{r=k}^n(2r+1)$

= $sum_{k=1}^n rac {1} {k} (sum_{r=1}^n(2r+1) - sum_{r=1}^{k-1}(2r+1))$

= $(n+1)^2 sum_{k=1}^n rac{1}{k} - sum_{k=1}^n rac{1}{k} k^2$

= $(n+1)^2 f(n) - rac{n(n+1)}{2}$

anchit saini

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6 Apr 2008 20:33:39 IST

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ohhh!!!

thank you sir

morale -->

one should post doubtful questions on goiit (hsbhatt sir) rather than looking into solutions of books :)

Anand Hegde

Blazing goIITian

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6 Apr 2008 20:34:45 IST

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source of the book anchit?

anchit saini

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6 Apr 2008 20:35:31 IST

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lol

source of the book is book shop !!!

Anand Hegde

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6 Apr 2008 20:50:18 IST

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lol... i meant source of the prob......

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