Series

I knew from my own reasoning, but I googled
it anyway. Does 1/n^.5 form a convergent or divergent
series... It is divergent! Below, an interesting page from
Dummies on the question of p-series. (In French, these are
called Riemann series).

https://www.dummies.com/education/math/calculus/how-to-recognize-a-p-series/

From the Wikipedia entry on mathematical series:

For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 19th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of the race, the tortoise has reached a second position; when he reaches this second position, the tortoise is at a third position, and so on. Zeno concluded that Achilles could never reach the tortoise, and thus that movement does not exist. Zeno divided the race into infinitely many sub-races, each requiring a finite amount of time, so that the total time for Achilles to catch the tortoise is given by a series. The resolution of the paradox is that, although the series has an infinite number of terms, it has a finite sum, which gives the time necessary for Achilles to catch up with the tortoise.

From the French-language entry:

L'étude des séries à termes réels ou complexes, sans hypothèse particulière, peut poser plus de problèmes. Une condition suffisante a une grande importance : si la série des valeurs absolues (série à termes réels) ou des modules (séries à termes complexes) ${\displaystyle \sum \left|a_{n}\right|}$ converge, alors la série ${\displaystyle \sum a_{n}}$ converge également. Elle est alors dite absolument convergente.

Il existe des séries convergentes sans être absolument convergentes, comme la série harmonique alternée ${\displaystyle \sum {\frac {(-1)^{n}}{n}}}$. Les méthodes d'étude pour ce type de série, plus techniques, (critère de convergence des séries alternées, théorème d'Abel, …) sont présentées dans l'article détaillé Série convergente.

From the German:

And - helpful - the Russian:

Animation montrant la convergence des sommes partielles d'une progression géométrique (ligne rouge) à son montant(ligne bleue) avec . Avec Maple.