Proving the Divergence of A+B Series

In summary, the conversation discusses proving the divergence of the sum of two series, A and B, if one series converges and the other diverges. The concept of convergence is determined by the row of partial sums and if both series converge, their sum will also converge. The conversation also mentions using Dimitri Terryn's suggestion and the rules of infinite series to prove the divergence of the sum of A and B.
  • #1
mrnoll
2
0
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
 
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  • #2
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.
 
  • #3
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)?
 
  • #4
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... :)
 
  • #5
it is that easy because what the infinite series converges to is defined to be the limit of the sequence of partial sums.
 
  • #6
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".
 

1) What is the A+B series and why is proving its divergence important?

The A+B series is a mathematical series where the terms alternate between two sequences A and B. Proving its divergence is important because it helps determine the behavior of the series and its limit, which has practical applications in fields such as engineering and economics.

2) How is the divergence of the A+B series proven?

The divergence of the A+B series can be proven using the limit comparison test or the ratio test. These tests compare the given series to a known divergent series and show that the given series also diverges.

3) Can the divergence of the A+B series be proven using the integral test?

No, the integral test cannot be used to prove the divergence of the A+B series. This is because the integral test can only be applied to series with non-negative terms, while the A+B series can have alternating positive and negative terms.

4) Is it possible for the A+B series to converge?

Yes, the A+B series can converge if the sequences A and B have the same convergence behavior and the series formed by their absolute values converges. In this case, the A+B series will also converge to the same limit.

5) What practical applications does the divergence of the A+B series have?

The divergence of the A+B series has practical applications in engineering, economics, and physics. It can be used to analyze the behavior of alternating current circuits, determine the stability of economic systems, and understand the behavior of oscillating systems in physics.

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