It is proved that if the series ∑ UN satisfies (1) Limun = 0, (2) ∑ (u2n-1 + u2n) convergence, then ∑ UN converges

It is proved that if the series ∑ UN satisfies (1) Limun = 0, (2) ∑ (u2n-1 + u2n) convergence, then ∑ UN converges

Reference example:
It is proved that if the positive series ∑ UN converges, then ∑ UN ^ α (α > 1) converges
answer:
∵limUn=0
lim(Un^a/un)=lim(un^(a-1))=0
If the positive series ∑ UN converges, then ∑ UN ^ α (α > 1) converges