# Limit of a Sequence

## Convergent and divergent sequences

Consider these sequences $$1,\frac{1}{2},\frac{1}{3},\cdots,\frac{1}{n},\cdots$$ $$1,3,5,7,\cdots,2n-1,\cdots$$ $$5,3,1,-1,\cdots,-2n+7,\cdots$$ $$-1,1,-1,\cdots,(-1)^n,\cdots$$ $$-1,2,-3,\cdots,(-1)^nn,\cdots$$ As $$n$$ gets larger and larger, let's see how the $$n$$th term changes.
$$a_{n}=\frac{1}{n}$$ As $$n$$ approaches infinity, You can see that the sequence $$a_{n}$$ gets closer to $$0$$

In a sequence $$\{a_{n}\}$$, as $$n$$ approaches infinity, if $$a_{n}$$ get closer to a certain value $$\alpha$$, we call the the sequence $$\{a_{n}\}$$ convergent, and we call $$\alpha$$ the limit of the sequence $$\{a_{n}\}$$. We represent it like this: $$\lim_{n \rightarrow \infty}a_{n}=\alpha$$ Note 1 In $$a_{n}=\frac{1}{n}$$, $$\lim_{n \rightarrow \infty}a_{n}=\lim_{n \rightarrow \infty}\frac{1}{n}=0$$
Note 2 In a sequence $$\{a_{n}\}$$ $$\alpha, \alpha, \cdots, \alpha, \cdots$$, we also say this sequence converges to $$\alpha$$, and we can represent it like this:$$\lim_{n \rightarrow \infty}a_{n}=\lim_{n \rightarrow \infty}\alpha=\alpha$$

$$b_{n}=2n-1$$ As $$n$$ approaches infinity, You can see that the sequence $$b_{n}$$ also approaches infinity. $$c_{n}=-2n+7$$ As $$n$$ approaches infinity, You can see that absolute value of the sequence $$c_{n}$$ approaches infinity.
We call these sequences divergent.
We can represent the limit of these sequences like this: $$\lim_{n \rightarrow \infty}b_{n}=\infty$$ $$\lim_{n \rightarrow \infty}c_{n}=-\infty$$
$$d_{n}=(-1)^n$$ $$e_{n}=(-1)^nn$$ In these sequences, as $$n$$ approaches infinity, the $$n$$th term neither converges nor approaches infinity (or negative infinity). Therefore, we call it divergent.

Note 3 We cannot write like this: $$\lim_{n \rightarrow \infty}(-1)^n=\pm{1}$$

## Properties

For any converging sequences $$\{a_{n}\},\{b_{n}\}$$, let $$\lim_{n \rightarrow \infty}a_{n}=\alpha, \lim_{n \rightarrow \infty}b_{n}=\beta$$ $$\lim_{n \rightarrow \infty}ka_{n}=k\alpha$$ $$\lim_{n \rightarrow \infty}(a_{n}\pm b_{n})=\alpha \pm\beta$$ $$\lim_{n \rightarrow \infty}a_{n}b_{n}=\alpha\beta$$ $$\lim_{n \rightarrow \infty}\frac{a_{n}}{b_{n}}=\frac{\alpha}{\beta} (b_{n}\ne 0, \beta\ne 0)$$ Keep in mind that these properties only work for converging sequences $$\{a_{n}\},\{b_{n}\}$$

## Example

1. Evaluate the following
a) $$\lim_{n \rightarrow \infty}\frac{5}{n}=\lim_{n \rightarrow \infty}\left(5\times \frac{1}{n}\right)=5\times 0=0$$
b) $$\lim_{n \rightarrow \infty}\frac{n+1}{n}=\lim_{n \rightarrow \infty}\left(1+ \frac{1}{n}\right)=1+0=1$$
c) $$\lim_{n \rightarrow \infty}\frac{n^2}{2n}=\lim_{n \rightarrow \infty}\frac{n}{2}=\infty$$

Use the fact that $$\lim_{n \rightarrow \infty}\frac{1}{n}=0$$

## Practice

1. Evaluate the following
a) $$\lim_{n \rightarrow \infty}\frac{2n^2-3n}{3n^2+2n-1}$$
b) $$\lim_{n \rightarrow \infty}\frac{n^2+3n}{2n-1}$$
c) $$\lim_{n \rightarrow \infty}\frac{n^2+1}{n^3-2n}$$
d) $$\lim_{n \rightarrow \infty}\left\{\log(n+2)-\log(n)\right\}$$

2. Evaluate the following
a) $$\lim_{n \rightarrow \infty}\frac{1^2+2^2+3^3+\cdots+n^2}{n^3+2}$$
b) $$\lim_{n \rightarrow \infty}\frac{1\times 2+2\times 3+3\times 4+\cdots+n(n+1)}{n(1+2+3+\cdots+n)}$$

## Limit of a geometric sequence

Convergence and divergence of geometric sequence $$r, r^2, r^3, \cdots, r^{n-1}, r^n, \cdots$$ depends on what value $$r$$ is.

Let's examine $$\lim_{n \rightarrow \infty}r^n$$
If $$r\gt 1$$, $$\lim_{n \rightarrow \infty}r^n=\infty$$
If $$r=1$$, $$\lim_{n \rightarrow \infty}r^n=1$$
If $$|r|\lt 1$$, $$\lim_{n \rightarrow \infty}r^n=0$$
If $$r=-1$$, $$\lim_{n \rightarrow \infty}r^n$$ does not exist
If $$r\lt-1$$, $$\lim_{n \rightarrow \infty}r^n$$ does not exist