suppose two series sum(ak) and sum(bk) such taht ak=bk for all but a
finite number of indices.  Show that if sum(ak) converges if and only
if sum(bk) converges

Hi fmunshi!!

Because Ak = Bk for all but a finite number of indices there is a
number p such that for all indices k > p is Ak = Bk.
Supose that sum(Ak) converges to A then:
sum(Bk)(k=1 to oo) = sum(Bk)(k=1 to p) + sum(Bk)(k=p+1 to oo) =
                   = sum(Bk)(k=1 to p) + sum(Ak)(k=p+1 to oo) =  
                   = sum(Bk)(k=1 to p) + (A - sum(Ak)(k=1 to p)) =
                   = A + [sum(Bk)(k=1 to p) - sum(Ak)(k=1 to p)] = 

If we call  R = [sum(Bk)(k=1 to p) - sum(Ak)(k=1 to p)], R is clearly
a finite number; then:
sum(Bk)(k=1 to oo) = A + R. 
That confirms the convergence of the serie sum(Bk) as a consequence of
the convergence of the serie sum(Ak).

Now supose that sum(Bk) converges to B then:
sum(Ak)(k=1 to oo) = sum(Ak)(k=1 to p) + sum(Ak)(k=p+1 to oo) =
                   = sum(Ak)(k=1 to p) + sum(Bk)(k=p+1 to oo) =  
                   = sum(Ak)(k=1 to p) + (B - sum(Bk)(k=1 to p)) =
                   = B - [sum(Bk)(k=1 to p) - sum(Ak)(k=1 to p)] = 
                   = B - R 
then:
sum(Ak)(k=1 to oo) = B - R. 

That confirms the convergence of the serie sum(Ak) as a consequence of
the convergence of the serie sum(Bk).

Then if two series sum(Ak) and sum(Bk) such taht Ak=Bk for all but a
finite number of indices, then sum(Ak) converges if and only if
sum(Bk) converges.

Hope this helps, if you find something obscure in this answer, please
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