Proving the Divergence of A+B Series

[mrnoll]

hi.

how can I prove that if A is a converges and B diverges

that the Sum of these series (A +B) diverges..



( A = a1 + a2 + a3 + ...
B = b1 + b2 + b3 + ...) if the series start from n=1

 

[Kalimaa23]

Limit of a sum is the sum of the limits.
The convergence of the series is determined by the row of partial sums.
Therefore, if both rows converge, the sum of both will too.

 

[HallsofIvy]

In order to apply Dimitri Terryn's suggestion, you will also have to note that if C=(A+ B), then Sum(B)= Sum(C- A). Assume both Sum(C) and Sum(A) converge. What does that tell you about Sum(B)?

 

[mrnoll]

Hi,

I don't think it´s that easy actually.
Because what I forgot to say was that these series go from
n=1 to n=infinity.

And in infinity normal rules don't always apply like SumA + SumB = Sum(A+B)

I need a proof... :)

 

[phoenixthoth]

it is that easy because what the infinite series converges to is defined to be the limit of the sequence of partial sums.

 

[HallsofIvy]

I don't think it´s that easy actually.
Because what I forgot to say was that these series go from
n=1 to n=infinity.

And in infinity normal rules don't always apply like SumA + SumB = Sum(A+B)

In this case they do. If Sum(A) and Sum(B) exist then
Sum(A+ B) exists and is equal to Sum(A)+ Sum(B).

I had assumed you were talking about infinite series since otherwise it wouldn't make sense to talk about "convergent" and "divergent".