Infinite series

An infinite series is a sum of an infinite sequence of numbers
Sums are determined with sequences by taking the limit of partial sums as the number of partial sums tends to infinity
Partial sums could be thought of as generating a new sequence where $a_{n} = s_{1} + ... s_{n}$ . You can analyse then what that sum approaches

Geometric series

Are of the form $a + a r + a r^{2} + ... + a r^{n - 1} + ... = \sum_{n = 1}^{\infty} a r^{n - 1}$
$a$ and $r$ are fixed real numbers and $a \neq = 0$
It doesn’t seem that r can be greater than 1?
If $r = 1$ the series diverges $lim_{n \to \infty} s_{n} = \pm \infty$ depending on the sign of $a$
If $r = - 1$ the partial sums alternate between $a$ and 0. This is a divergence.
If $∣ r ∣ \neq = 1$ we determine convergence or divergence using
$s_{n} = \frac{a ( 1 - r ^{n} )}{1 - r}, (r \neq = 1)$
Applies only if the n index starts at 0 or 1 with n-1 as the power
I’ve worked through this on paper. When $∣ r ∣ < 1$ $r^{n}$ will tend towards 0 so the series converges. If $∣ r ∣ > 1$ the partial sum will tend towards positive or negative infinity so it diverges

If $\sum_{n = 1}^{\infty} a_{n}$ converges, then $a_{n} \to 0$
If the integral converges or diverges, so does the series. For a sequence of positive terms $a_{n}$ , if $a_{n} = f (n)$ where $f$ is continuous and positive and decreasing, that is for each $x_{1} \l x_{2}$ in the domain $f (x_{1}) \geq f (x_{2})$

ratio test if we’ve a series $\sum a_{n}$ and $l i m_{n \to \infty} ∣ \frac{a _{n + 1}}{a _{n}} ∣ = ρ$ , the series converges absolutely if $ρ < 1$ , diverges if $ρ > 1$ and is inconclusive if $ρ = 1$
Going to have to take for granted that we’ve a bunch of tests for convergence of a series

$\sum_{n = 0}^{\infty} c_{n} (x - a)^{n} = c_{o} + c_{1} (x - a) + ... + c_{n} (x - a)^{n} + ...$
If we take all the constants to be 1, you get the geometric series that converges to $\frac{1}{1 - x}$ (from above)
If we shift focus to think of a partial sum of highest polynomial n as $P_{n} (x)$
If $a_{n}$ (being the partial sum) is on the y axis and n (or the index of the series) along the x axis. Graphically, each increase along the x is trying to get closer to some incredibly large y value.
The above has a boundary of plus and minus 1. As x gets closer and closer to 1, $P_{n} (x)$ approaches the sum $\frac{1}{1 - x}$
power series behaves in 3 possible ways
- It might converge at a single value
- converge everywhere
- converge on some interval, this is like the bounds of -1 and 1 above right?

If we take the sum of a power series as a function $f (x) = \sum_{n = 0}^{\infty} a_{n} (x - a)^{n}$
There are an infinite number of derivates in the interval of convergence
If we had a set of these intervals, could we reconstruct the series, or at least parts of it, in turn, approximating a function based on the power series.

Is the significance of polynomials in the derivative nature, that it intuitively makes sense that they’re approximating a function as each higher order of polynomial means another (more specific) derivative on the initial convergence interval?
Also, watched a video that details how, similar to converting fractions to decimal places, making addition easier. Converting functions to approximations makes addition easier.