Cauchy sequence and sums of distances of consecutive terms

Question

Consider a sequence $(x_n)$ in metric space $(X,d)$.

If the sequence of partial sums $\sum_{i=1}^n d(x_i,x_{i+1})$ of distances between consecutive terms converges in $\mathbb{R}$, then I understand why the sequence is Cauchy.

But is the converse statement true as well?

Answer 1

No. A counterexample is afforded by the sequence $a_n=(-1)^n/n$ in $\mathbb R$ with the standard metric, which is Cauchy but for which the sum of distances is essentially twice the harmonic series and thus diverges.