From Pythagoras to Fourier and From Geometry to Nature

DOI: https://doi.org/10.55060/b.p2fg2n.ch002.220215.005

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We recall, in a schematic way, the basic definitions about sequences and series [62].

A **sequence** is a function defined on the set **N** of integers (*n* ∈ **N**).

A property is said to be *definitively* true if it holds for all integer indices greater than a given value.

The **sequence**

The **sequence** *n* diverges positively. We write: *n* → +∞.

The **sequence** {(−1)^{n}} is indeterminate: it does not admit a limit.

The **sequence** {*a _{n}*} converges to the limit

The **sequence** {*a _{n}*} is convergent if and only if ∀

The **sequence** {*a _{n}*} diverges positively ⇔ ∀

A **series** is the sum of the terms of a sequence:

The study of **series** reduces to that of the **sequence** of partial sums:

If {*s _{n}*} →

If {*s _{n}*} → +∞ the

If {*s _{n}*} does not have a limit the

The **series** of Zeno’s paradox is convergent:

The harmonic **series** is positively divergent:

The **series**

If {*a _{n}*} depends on

For example, the **series** of functions *S*(*x*) in (*a*,* b*) if ∀*x* ∈ (*a*,* b*) we have:

The convergence of this **series** is uniform in [*α*,* β*] ⊂ (*a*,* b*) if:

It is proven that the limit of a uniformly convergent **sequence** of continuous functions is a continuous function.

Example: consider the geometric **series**

Recalling the equation 1 − *x ^{n}* = (1 −

Since:

Moreover, the convergence cannot be uniform in [−1, 1], but only in intervals of the type [*α*,* β*] ⊂ (−1, 1).