# can someone prove this

• April 25th 2006, 12:20 PM
miss_lolitta
can someone prove this
Hello,
can someone prove this to me as.
Any help would help save thier hair I have not torn out as yet. :mad:

If
$a_n,b_n$
are sequences of real number ,n>m then:

$\sum_{k=m}^{n}a_k .b_k=$ $a_{n+1}S_n-a_m S_{m-1}+\sum_{k=m}^{n}( a_k - b_{k+1})S_k$

Where
$S_n$
is the partial sum of sequence
$\sum_{k=1}^{\infty}b_n$

Thanks for any help :)
• April 25th 2006, 02:12 PM
CaptainBlack
Quote:

Originally Posted by miss_lolitta
Hello,
can someone prove this to me as.
Any help would help save thier hair I have not torn out as yet. :mad:

If
$a_n,b_n$
are sequences of real number ,n>m then:

$\sum_{k=m}^{n}a_k .b_k=$ $a_{n+1}S_n-a_m S_{m-1}+\sum_{k=m}^{n}( a_k - b_{k+1})S_k$

Where
$S_n$
is the partial sum of sequence
$\sum_{k=1}^{\infty}b_n$

Thanks for any help :)

Either my correction of the TeX has introduced an error, or this cannot
be an identity, as the LHS is independent of $a_{n+1}$, while the RHS is dependent
on $a_{n+1}$.

RonL
• April 27th 2006, 07:28 AM
miss_lolitta
the relation is correct..sure
• April 29th 2006, 12:22 PM
miss_lolitta
Were is the proof??
• April 29th 2006, 12:25 PM
CaptainBlack
Quote:

Originally Posted by miss_lolitta
Were is the proof??

There is no proof because it cant be an identity.

RonL
• April 29th 2006, 12:29 PM
miss_lolitta
How??
• April 29th 2006, 01:25 PM
CaptainBlack
Quote:

Originally Posted by miss_lolitta
How??

We seek to prove that for arbitary sequences $a_i,\ i=1 \dots$ and $b_i,\ i=1 \dots$,
and $S_n$ the sequence of partial sums of the $b_i$'s:

$\sum_{k=m}^{n}a_k .b_k=$ $a_{n+1}S_n-a_m S_{m-1}+\sum_{k=m}^{n}( a_k - b_{k+1})S_k\ \ \dots(1)$.

Rearranging:

$a_{n+1}=(-\sum_{k=m}^{n}a_k .b_k+$ $a_{n+1}S_n-a_m S_{m-1}+\sum_{k=m}^{n}( a_k - b_{k+1})S_k\left)/S_n$

Now everything on the RHS is independent of $a_{n+1}$, so
$a_{n+1}$ is deternined by the sequence $b_i,\ i=1\dots$
and $a_i,\ i=1 \dots n$, and so $a_i,\ i=1 \dots$ is not an arbitary sequence,
or if it is then $(1)$ is not an identity.

RonL