From Pythagoras to Fourier and From Geometry to Nature

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

Chapter 2. Recalling Sequences & Series

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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 (nN).

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

The sequence 1n converges to zero. We write: 1n0.

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

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

The sequence {an} converges to the limit ⇔ |an| → 0.

The sequence {an} is convergent if and only if ∀ɛ > 0 it results definitively that |anam| < ɛ (Cauchy’s convergence criterion).

The sequence {an} diverges positively ⇔ ∀K > 0 it results definitively that |an| > K.

A series is the sum of the terms of a sequence: k=1ak.

The study of series reduces to that of the sequence of partial sums: sn=k=1nak.

If {sn} → s the series is convergent and its sum is s.

If {sn} → +∞ the series diverges positively.

If {sn} does not have a limit the series is said to be indeterminate.

The series of Zeno’s paradox is convergent: k=01/2n=2.

The harmonic series is positively divergent: k=11/k=+.

The series k=0(1)k is indeterminate.

If {an} depends on x ∈ (a, b) we have sequences or series of functions.

For example, the series of functions k=1ak(x) converges to S(x) in (a, b) if ∀x ∈ (a, b) we have:

|Sn(x)S(x)|=|k=1nak(x)S(x)|0

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

maxx[α,β]|Sn(x)S(x)|0

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

Example: consider the geometric series k=0xk .

Recalling the equation 1 − xn = (1 − x)(1 + x + x 2 + ⋯ + xn−1), assuming x ≠ 1, it follows that:

Sn(x)=1+x+x2++xn1=1xn1x

Since:

xn{0if|x|<1+ifx>1indeterminateifx1
we can conclude that:
k=0xk={11xif|x|<1+ifx1indeterminateifx1

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