# Conceptual question regarding sum of series

• fk378
In summary, the discussion revolves around whether the sum of two divergent series, a_n and b_n, will necessarily result in a divergent sum of (a_n + b_n). It is concluded that this is not always the case, as there are examples where the sum of (a_n + b_n) is 0 and therefore convergent.
fk378

## Homework Statement

If the sum(a_n) and the sum(b_n) are both divergent, is the sum(a_n + b_n) necessarily divergent?

## The Attempt at a Solution

At first I thought it must be divergent, but then I asked, what if a_n is 1/n and b_n is -1/n... then their sum would be 0.

Does this logic make sense? And if so, would it imply that sum(a_n + b_n)=0, and is therefore convergent?

Your logic is correct. We can even get simpler examples. a_n = n, b_n = -n. Each diverges individually, but a_n + b_n = 0 for every term. And yes, it implies the sum of (a_n + b_n) is 0, convergent.

## 1. What is a "sum of series" in mathematics?

A sum of series is a mathematical concept that involves adding together a sequence of numbers or terms in a specific order. The result of this addition is called the "sum" of the series.

## 2. How do you calculate the sum of a series?

The sum of a series can be calculated by adding together all the terms in the series. This can be done manually by hand or with the use of a calculator. There are also specific formulas and methods for calculating the sum of different types of series, such as arithmetic and geometric series.

## 3. What is the difference between an infinite series and a finite series?

An infinite series is a series that has an infinite number of terms, meaning it does not have an end point. A finite series, on the other hand, has a specific number of terms and therefore has an end point. Calculating the sum of an infinite series can be more complex than a finite series, as the number of terms to be added is infinite.

## 4. Can the sum of a series be negative?

Yes, the sum of a series can be negative. This can happen when the individual terms in the series alternate between positive and negative values, resulting in a net negative sum. Additionally, some series have a negative value as the starting term, which can also lead to a negative sum.

## 5. What is the significance of the sum of a series in mathematics?

The sum of a series plays an important role in various mathematical concepts and applications. It is used to find the total value of a sequence of numbers, and can also be used to represent real-world situations such as compound interest or population growth. Additionally, the sum of a series is often used in calculus and other advanced mathematical fields.

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