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I've seen the phrase 'digits per term', mostly with regards to algorithms that produce $\pi$. I've seen it here ("Yes, Chudnovsky's formula converges at a steady 14.18 digits per term."), on numerous formulas for $\pi$ ("This gives 50 digits per term", "...converges at only one bit/term") and on source code involving Khinchin's constant. I've fiddled with one of the algorithms, and it appears that it's -log(value of ratio test)/log(10), but I don't know why it's that value nor if it would be true for any method.

The phrase itself implies a linear convergent series, but could it be used on any converging series? It also seems to be more than just saying a series converges linearly or quadratically. For instance, given a series, could I say something like "it produces $≈0.7503$ digits per term" or "$0.023n^2$ correct digits per $n^{th}$ term are found"?

Just what exactly is it (if there is a definition for it) and how is it determined? Just where are these numbers coming from? Searching for said phrase just leads me to stuff about $\pi$.

Status
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  • So what don't you understand in the context of those particular references? As far as I can see, those references are all about series approximations and "digits [or bits] per term" makes good sense. I agree that you might be working series which converge too slowly or too quickly for this to be a useful measure, so you might want to say that the $n$-th term adds $f(n)$ digits of additional precision for some non-constant function $f$. I don't see why you think the concept of the number of digits of precision added by each term in a series needs any more detailed definition. – Rob Arthan Feb 07 '19 at 20:10
  • @RobArthan Where do these numbers even come from? – Status Feb 07 '19 at 21:31

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