# Series Convergence: Show AK -> 0 as K->∞

• hth
In summary, the conversation discusses a proof that shows if a series \sumak is convergent, then the sum from k to infinity of ak goes to zero as k goes to infinity. The attempt at a solution involves showing that \sumak is convergent using the definition of convergence, and then using the definition of convergence of a series to find the sum from k to infinity in terms of a partial sum.
hth

## Homework Statement

Show that if $$\sum$$ak converges, then $$\sum$$ from k to ∞ of ak goes to zero as k goes to ∞.

## The Attempt at a Solution

First I tried to show that $$\sum$$ak is convergent.

Let c be a real number and ε > 0. So there is an integer N > 0 such that if n > N then |an - c | < ε.

So c is the limit of the sequence and an -> c.

I don't really know where to go from there. Any help is appreciated.

The definition of convergence of a series uses partial sums. What's the sum from k to infinity in terms of a partial sum?

## 1. What is the concept of series convergence?

Series convergence is a mathematical concept that refers to the behavior of a sequence of numbers as the number of terms in the sequence increases. It involves determining whether the sum of an infinite series of numbers approaches a finite value as the number of terms increases, or whether it approaches infinity.

## 2. How can we show that a series converges to 0?

To show that a series converges to 0, we need to prove that the terms in the series become smaller and smaller as the number of terms increases, eventually approaching 0. This can be done by using mathematical techniques such as the comparison test, limit comparison test, or ratio test.

## 3. What is the importance of proving AK -> 0 as K->∞ in series convergence?

Proving AK -> 0 as K->∞ is important in series convergence because it shows that the series approaches a finite value or converges to 0. This information is crucial in determining the convergence or divergence of a series and helps in solving various mathematical problems and applications.

## 4. What are some common methods used to prove AK -> 0 as K->∞?

Some common methods used to prove AK -> 0 as K->∞ include the comparison test, limit comparison test, ratio test, root test, and integral test. Each of these methods has its own specific conditions and can be used depending on the type of series and its terms.

## 5. How is the concept of series convergence applied in real life?

The concept of series convergence has various applications in real life, such as in physics, engineering, finance, and computer science. It is used to determine the behavior of systems and models, analyze data, and make predictions. For example, it is used in calculating the value of investments, estimating the strength of materials, and designing efficient algorithms.

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