Theory

Theory 1

A sequence has a limit if its terms tend toward a specific number, or toward $\pm \infty$.

When this happens we can write “${\lim }_{n\to \infty }{a}_n=L$” with some number $L\in ℝ$ or $L=\pm \infty$. We can also write “$a_n\to L$ as $n\to \infty$”.

The sequence is said to converge if it has a finite limit $L\in ℝ$.

Some sequences don’t have a limit at all, like $a_n=\cos n$:

Or $a_n=e^n$:

These sequences diverge. In the second case, there is a limit $L=\infty$, so we say it diverges to $+\infty$.

A sequence may have a limit of $\pm \infty$ but is still said to diverge.

Extra - Convergence definition

The precise meaning of convergence is this. We have $a_n\to L$ as $n\to \infty$ if, given any proposed error $\epsilon >0$, it is possible to find $N$ such that for all $n>N$ we have $|a_n-L|<\epsilon$.

When $L=\infty$, convergence means that given any $B>0$, we can find $N$ such that for all $n>N$ we have $a_n>B$.

Similarly for $L=-\infty$.

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If the general term $a_n$ is a continuous function of $n$, we can replace $n$ with the continuous variable $x$ and compute the continuous limit instead:

$$\underset{n\to \infty }{\lim }{a}_n=\underset{x\to \infty }{\lim }{a}_x$$

If $a_x$ would be a differentiable function, and we discover an indeterminate form, then we can apply L’Hopital’s Rule to find the limit value. For example, if the indeterminate form is $0\cdot \infty$, we can convert it to $\frac{\infty }{1/0}=\frac{\infty }{\infty }$ and apply L’Hopital.

Theory 2

Monotone sequences

A sequence is called monotone increasing if $a_{n+1}\ge a_n$ for every $n$.

A sequence is called monotone decreasing if $a_{n+1}\le a_n$ for every $n$.

In this context, ‘monotone’ just means it preserves the increasing or decreasing modality for all terms.

Monotonicity Theorem

If a sequence is monotone increasing, and bounded above by $B$, then it must converge to some limit $L$, and $L\le B$.

If a sequence is monotone decreasing, and bounded below by $B$, then it must converge to some limit $L$, and $L\ge B$.

Terminology:

• Bounded above by $B$ means that $a_n\le B$ for every $n$
• Bounded below by $B$ means that $B\le a_n$ for every $n$

Notice!

The Monotonicity Theorem says that a limit $L$ exists, but it does not provide the limit value.

Theory 3

Series convergence

We say that a series converges when its partial sum sequence converges:

$$\text{“}\sum _{n=0}^{\infty }{a}_n\text{converges}\text{”}\text{MEANS:}\text{“}{S}_N\text{converges as }N\to \infty \text{”}$$

Let us apply this to the geometric series. Recall our formula for the partial sums:

$$S_N=a_0\frac{1-r^{N+1}}{1-r}$$

Rewrite this formula:

$$\gg \gg S_N=\frac{a_0}{1-r}-\frac{a_0}{1-r}{r}^{N+1}$$

Now take the limit as $N\to \infty$:

$$\underset{N\to \infty }{\lim }{S}_N=\text{“}\frac{a_0}{1-r}-\frac{a_0}{1-r}{r}^{\infty +1}\text{”}=\frac{a_0}{1-r}$$

So we see that $S_N$ converges exactly when $|r|<1$. It converges to $\frac{a_0}{1-r}$.

(If $|r|=1$ then the denominator is $0$, and if $|r|>1$ then the factor $r^{\infty +1}$ does not converge.)

Furthermore, we have the limit value:

$$\sum _{n=0}^{\infty }{a}_n\gg \gg \underset{N\to \infty }{\lim }{S}_N\gg \gg \frac{a_0}{1-r}\gg \gg S$$

This result confirms the formula we derived for the total sum $S$ of a geometric series. This time we did not start by assuming $S$ exists, rather we proved that $S$ exists. (Provided that $|r|<1$.)

Extra - Looking at $S$ and $S_N$

Notice that we always have the rule:

$$\begin{array}{cc}{S}_N& =S-r^{N+1}S\\ & \\ S_N& =\frac{a_0}{1-r}-\frac{a_0}{1-r}{r}^{N+1}\end{array}$$

This rule can be viewed as coming from partitioning the full series into a finite part $S_N$ and the remaining infinite part:

$$$$

We can remove a factor $r^{N+1}$ from the infinite part:

$$S-S_N=r^{N+1}(a_0+a_{0}r+a_0{r}^2\dots )$$

The parenthetical expression is equal to $S$, so we have the formula $S_N=S-r^{N+1}S$ given above.