Does the definition of convergence of an infinite series go both ways?

Question

We say a series converges if the sequence of partial sums converges. That seems like it is a definition, and not a theorem. So if I am given an infinite series, and I know it converges, am I allowed to say the sequence of partial sums converges? Or if I know the sequence of partial sums converges, may I say the series converges?

Answer 1

In mathematics, a concept $p$ is defined using another concept $q$, by saying $p$ holds if $q$ holds.
This is written as $p:=q$.
In logic, this means $p\iff q$, i.e., both are equivalent ways of referring to the same concept defined.

In a concrete example, as in your post,
Suppose $(x_n)_{n\ge1}$ is a sequence. Corresponding to this, define a sequence of partial sums $(\sum_{k=1}^n x_k)_{n\ge1}$

The series $\sum_{n=1}^{\infty} x_n$ is convergent $:=$ the sequence $(\sum_{k=1}^n x_k)_{n\ge1}$ converges,

which means $\sum_{n=1}^{\infty} x_n$ is convergent $\iff$ $(\sum_{k=1}^n x_k)_{n\ge1}$ is convergent.