*Now* calculus rears its not-so-ugly head. Sequences are like bulls at a rodeo waiting to be lassoed, but the divergent ones can't be caught. Let's make sure we are comfortable with limits, and let's see which sequences we can stop.

In some of the sequences we were graphed, it looked like as *n* got bigger the values *a*_{n} approached some particular value.

### Sample Problem

The terms of the sequence approach 0 as *n* approaches ∞.

When the terms of a sequence approach some finite value *L* as *n* gets bigger, we say the sequence *converges *to *L*, as is lasso. In symbols, a sequence converges to *L* if . This is just like convergence for functions. These are the bulls we can wrangle.

### Sample Problem

The sequence converges to 1, because

If we look at a convergent sequence on a number line, it looks like the dots are getting closer and closer to value *L*.

### Sample Problem

The sequence converges to 0. The dots are trying to get to 0 on the number line.

If we look at a convergent sequence on a 2-D graph, it looks like a function with a horizontal asymptote. The dots will get closer and closer to height *L* as *n* gets bigger.

### Sample Problem

The sequence converges to 1. As *n* gets bigger and we move to the right on the graph, the dots get closer and closer to height 1.

Just like a function, if a sequence doesn't converge, we say it *diverges*. Sequences can diverge for different reasons.

### Sample Problem

The sequence *a*_{n} = *n* diverges because as *n* approaches ∞ the terms *a*_{n} approach ∞ also. Since the terms aren't getting closer to anything finite, we say the sequence diverges. This is the bull that gets away from you before you can lasso it.

### Sample Problem

The sequence *a*_{n} = (-1)^{n} diverges because it's indecisive and can't make up its mind whether to be + 1 or -1. This is the bull that catches you and throws you over it's head using its horns. You'll get up, try to wrangle it, and it'll just throw you over its head again.

To determine if a sequence converges or diverges, see if the limit

exists and is finite. For the sake of intuition, it may be helpful to graph the sequence.

## Practice:

Determine if the sequence converges or diverges. If the sequence converges, what does it converge to?

Answer

This sequence is like except there's even more stuff in the denominator. This sequence converges to 0.

Determine if the sequence converges or diverges. If the sequence converges, what does it converge to?

*a*_{n} = 2^{n}

Answer

This sequence diverges. As *n* approaches ∞, the terms get farther and farther apart.

Determine if the sequence converges or diverges. If the sequence converges, what does it converge to?

*a*_{n} = 7

Answer

Since the terms are all 7, this sequence converges to 7.

Determine if the sequence converges or diverges. If the sequence converges, what does it converge to?

(hint: expand the factorial)

Answer

Following the hint, the *n*^{th} term of this sequence looks like

The factors in the denominator cancel with factors in the numerator, leaving

As *n* approaches ∞, the terms *a*_{n} also approach ∞, so this sequence diverges.

Determine if the sequence converges or diverges. If the sequence converges, what does it converge to?

Answer

The terms of this sequence bounce back and forth, getting farther away from zero. Since the terms aren't approaching any finite value, the sequence diverges.