If `|b_(k+1)//b_k|` approaches a limit greater than `1` — or diverges to infinity — then from some point on, `|b_(k+1)|` is always bigger than `|b_k|`. This means the terms `b_k` can't decrease in size to `0`, so the series must diverge.
For both the harmonic series and the alternating harmonic series, we have
`lim_(k rarr oo)|b_(k+1)/b_k|=lim_(k rarr oo)[1text[/(]k+1text[)]]/[1//k]=lim_(k rarr oo)k/(k+1)=1`.
But the harmonic series diverges, and the alternating harmonic series converges. Thus we cannot conclude anything about convergence of a series from the observation that
`lim_(k rarr oo)|b_(k+1)/b_k|=1`.