This is an infinite series:
[k, ∞] ∑a(k) = a1 + a2 + a3 .. + a∞
Sorry for the type. Here:
Here is a basic review of sequences and series. The sum of infinite geometric series will be incredibly important~
The bulk of investigating infinite series is about the power series. This is merely the sum of infinite geometric series, previously notated as [1, ∞] ∑ar^(n-1). Evaluating this sum gives c/(1-x), previously known as a/(1-r). You can have "x centered at a", notated as "x = a". The term "a" is essentially your x shift from the origin. We will come to understand the purpose later.
There are some fancy tricks you can do with the power series.. differentiate and integrate~ You can apply it to the notation itself, or expand a finite sum into its additions and differentiate / integrate term by term as you would a very long function.
The power series is all very well for polynomials and geometric sums, but for others we need to use the Taylor series:
This is merely the power series, having cn replaced with "the k order derivative of f(x) divided by k factorial". The purpose of all this can be better understood with a graph:
Say f(x) = e^x like in this graph. Pn(x) denotes an n degree Taylor polynomial, or a partial sum of f(x), or an approximation of f(x). An infinite sum gives you f(x), so the greater the n of Pn(x), the closer it approximates f(x). The series represented in this graph is centered at zero so that Pn(x) hugs f(x) at x = 0 (any series centered at zero is a Maclaurin series). It can be centered elsewhere to approximate some value far from the y axis. You might want to revisit power series with this in mind~
Since the polynomial Pn(x) is only a finite segment of the infinite function f(x), the excluded part makes the truncation error. The remainder Rn(x) of Pn(x) is the next order term. To put a number to it, the remainder estimation replaces f^(n+1)(c) with a larger version Mr^(n+1) that covers the interval [x, a].
Series and functions clash - funeries~
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