It is asked if it is convergent or Divergent series
1/(1*2) + 2/(3*4) + 3/(5*6)... infinity
2026-03-28 11:53:10.1774698790
Converging or diverging
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No; it diverges. First of all, note that $\dfrac n{n(n+1)}=\dfrac1{n+1}$. On the other hand, the series $\sum_{n=0}^\infty\frac1{n+1}$ is the harmonic series, which diverges.
The series that you wrote after editing your question is $\displaystyle\sum_{n=1}^\infty\frac n{(2n-1)2n}=\sum_{n=1}^\infty\frac1{4n-2}$ which diverges too, since $(\forall n\in\mathbb N):\dfrac1{4n-2}\geqslant\dfrac14\times\dfrac1n$ and since the harmonic series diverges.