If t_{k} is an infinite sequence, then an expression of the form:
t_{1} + t_{2} + t_{3} + \dots
it is called an infinite series or just a series. A series is the result of adding the elements that make up a sequence.
A series is an element of the sequence of partial sums
\begin{array}{l} S_{1} = t_{1}\\ S_{2} = t_{1} + t_{2} = S_{1} + t_{2}\\ S_{3} = t_{1} + t_{2} + t_{3} = S_{2} + t_{3} \\ \dots \\ S_{n} = t_{1} + t_{2} + t_{3} + \dots + t_{n} = S_{n-1} + t_{n} \end{array}Represented by the following expression:
S_{n} = S_{1}, S_{2}, S_{3}, \dots , S_{n}, \dots
Which we have the recursive formula:
S_{n} = S_{n-1} + t_{n} \ \text{ con } \ S_{1} = t_{1} \ \text{ y } \ k = 2, 3, 4, \dots
Let us remember that if the sequence that originates a series is convergent, then the sequence will also be convergent.
Series types
There are two types of series that are handled in numerical methods:
- Power series of x
\displaystyle \sum_{i=0}^{\infty}C_{i}x^{i} = C_{0} + C_{1}x + C_{2}x^{2} + \dots + C_{k}x^{k}
- Power series of \left( x-a \right)
\displaystyle \sum_{i=0}^{\infty}C_{i}\left( x - a \right)^{i} = C_{0} + C_{1}\left(x - a \right) + C_{2}\left( x - a\right)^{2} + \dots + C_{k}\left(x - a \right)^{k}
Applying a set of mathematical operations that are to equalize the series to a function and applying derivatives with respect to x up to the N-th and then evaluating the function and the derivatives at the point x=0, we will obtain that the x power series are the famous McLaurin series and the \left( x-a \right) power series are the famous Taylor series:
Power series of x = MacLaurin Series
f(x) = f(0) + f\text{'}(0)x + \cfrac{f\text{''}(0)x^{2}}{2!} + \cfrac{f\text{'''}(0)x^{3}}{3!} + \dots + \cfrac{f^{k}(0)x^{k}}{k!}
Power series of \left( x-a\right) = Taylor series
f(x) = f(a) + f\text{'}(a)\left(x - a\right) + \cfrac{f\text{''}(a)\left(x - a \right)^{2}}{2!} + \cfrac{f\text{'''}(a)\left( x - a \right)^{3}}{3!} + \dots
+ \cfrac{f^{k}(a)\left(x - a\right)^{k}}{k!}
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