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 1n converges to zero. We write: 1n→0.

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 ℓ ⇔ |a_n − ℓ| → 0.

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

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

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

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

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

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

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

The series of Zeno’s paradox is convergent: ∑k=0∞1/2n=2.

The harmonic series is positively divergent: ∑k=1∞1/k=+∞.

The series ∑k=0∞(−1)k is indeterminate.

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

For example, the series of functions ∑k=1∞ak(x) converges to 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 ∑k=0∞xk .

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

Since:

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