For the exponential series, we find that
`|b_(k+1)/b_k|=[k!|x|^(k+1)]/[text[(]k+1text[)]!|x|^k]=|x|/(k+1)`.
For `k=n`, this is precisely the ratio of terms in the geometric series we use to estimate the `n`th tail. Since `x` is fixed — and independent of `k` — the limiting value of the ratios is `0`, no matter how big `x` is. As we saw already, for any `k>text[|] x text[|]`, all the ratios will be less than `1` from then on, so the tail has a finite sum. Specifically, we can take `r` in the geometric series to be `text[|] x text[| / (|] x text[|] +1 text[)]`.
For the logarithmic case,
`|b_(k+1)/b_k|=[k|x|^(k+1)]/[text[(]k+1text[)]|x|^k]= k/(k+1)|x|`.
Recall that in this case we took `r` to be `text[|] x text[|]`. What we see from our ratio calculation is that the ratio of consecutive terms is always smaller than `text[|] x text[|]`. Thus we can be sure of convergence if `text[|] x text[|] <1`. We find that the limiting value of the ratios is also `text[|] x text[|]`, which means we can't expect these ratios to stay less than `1` unless `text[|] x text[|] <1`.