This is not true. The infinite series
[tex]\displaystyle\sum_{n=1}^\infty n[/tex]
converges if and only if the sequence of its partial sums converges. The [tex]k[/tex]-th partial sum is
[tex]\displaystyle\sum_{n=1}^kn=\frac{k(k+1)}2[/tex]
but clearly this diverges as [tex]k[/tex] gets arbitrarily large.
