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can someone prove this to me as.

Any help would help save my hair I have not torn out as yet.

If \(\displaystyle

a_n,b_n

\)are sequences of real number ,n>m then:

\(\displaystyle

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

\)

Where \(\displaystyle

S_n

\)is the partial sum of sequence \(\displaystyle

\sum_{k=1}^{\infty}b_n

\)

Thanks for any help