# How to determine convergence and divergence

• MHrtz
In summary, the problem is that you reason as follows: (1+2/n) goes to 1, n goes to infinity, so (1+2/n)n goes to 1^{+\infty}=1. The problem is however that 1^{+\infty} is an undetermined form and does not equal 1! Check your calculator, if you type in large values of n, then you'll see that (1+2/n)n does not go to 1!
MHrtz
I've been having some trouble understanding how to determine if a sequence is divergent or convergent. For example

an = cos(2/n)

I know if I take the limit as n ->$\infty$ then I will get 1. So the sequence has a limit but does having a limit mean that the sequence is convergent.

Hi MHrtz!

Yes, saying that a sequence has a limit (which should not be infinity) is equivalent to saying that the sequence converges.
If the limit doesn't exist (or is infinite), then the sequence diverges.

an {1 n = 2k k is an integer
{0 otherwise

It's divergent only sometimes right?

MHrtz said:

an {1 n = 2k k is an integer
{0 otherwise

It's divergent only sometimes right?

A sequence is either convergent or divergent. There's no such thing as divergent "only sometimes". This particular sequence is divergent since it doesn't come close to any number.

Ok, so i did some more problems and came across this one:

an = (1 + 2/n)n

When I took the limit I though it was 1 but the book said that the limit was e2. How is this possible?

The problem is that you probably reason as follows:

(1+2/n) goes to 1, n goes to infinity, so (1+2/n)n goes to $1^{+\infty}=1$. The problem is however that $1^{+\infty}$ is an undetermined form and does not equal 1! Check your calculator, if you type in large values of n, then you'll see that (1+2/n)n does not go to 1!

How to solve this problem then. Well, it depends on what you seen.
Some students have $(1+1/n)^n\rightarrow e$ as a separate formula. Then you just need to transform (1+2/n)n into something of the form (1+1/m)m (hint: m=n/2)

If you haven't seen that separate formula, then you can always do

$$(1+2/n)^n=e^{n\log(1+2/n)}$$

so you just need to show that

$$n\log(1+2/n)\rightarrow 2$$

## 1. What is convergence and divergence?

Convergence and divergence refer to the behavior of a mathematical sequence or series. Convergence means that the terms of the sequence or series approach a specific limit as the number of terms increases. Divergence means that the terms of the sequence or series do not approach a limit and instead continue to increase or decrease without bound.

## 2. How can you determine if a sequence is convergent or divergent?

One way to determine convergence or divergence is to calculate the limit of the sequence. If the limit exists and is a finite number, then the sequence is convergent. If the limit does not exist or is infinite, then the sequence is divergent. Additionally, you can also use various convergence tests, such as the ratio test or the integral test, to determine convergence or divergence.

## 3. What are some common types of series that are convergent or divergent?

Some common types of series that are convergent include geometric series, telescoping series, and p-series (series of the form 1/n^p where p > 1). Some common types of series that are divergent include harmonic series (series of the form 1/n) and alternating series where the terms do not decrease in absolute value.

## 4. How can you use the graph of a function to determine convergence or divergence?

If the graph of a function approaches a horizontal line (i.e. a constant value) as the input variable increases, then the series represented by that function is convergent. If the graph of a function continues to increase or decrease without approaching a horizontal line, then the series is divergent.

## 5. Why is it important to determine convergence and divergence?

Knowing whether a sequence or series is convergent or divergent is important in many areas of mathematics, including calculus, differential equations, and numerical analysis. It allows us to make accurate predictions and conclusions based on mathematical models and helps us understand the behavior of mathematical functions and equations.

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