How does truncating a series affect "upstream" values in the series?

Question

It is known that truncating the Gregory’s series to 5,000,000 terms leads to an "almost but not quite" value for π:

$$ \pi=4 \sum_{k=1}^{\infty} \frac{(-1)^{k+1}}{2 k-1}=4(1-1 / 3+1 / 5-1 / 7+\cdots) $$

Gregory’s series : 3.14159245358979323846464338327950278419716939938730582097494182230781640...
π :                3.14159265358979323846264338327950288419716939937510582097494459230781640...

Note the 7th decimal place is a 4 instead of the proper 6.

But how does truncation affect a number so far upstream from the truncation? In other words, if we are capping the series at 5 million decimal places, why would the 7th decimal place be affected? Is it a rounding that "propagates" back up the number?

UPDATE

It turns out that small perturbations of the correct decimal expansion for π appear when truncating the formula. It's actually not too surprising in terms of the problem of induction, however why the perturbations appear so early in the expansion is still quite interesting. In this particular case, capping the series at 5,000,000 steps ends up being exactly one-half of a fairly large power of ten, which apparently explains the perturbation, although I can't say I quite understand why.

Answer 1

You have just subtracted $\frac{4}{9,999,999}$ so your partial sum is too small by about half that

and are about to add $\frac{4}{10,000,001}$ which would make the partial sum too big by about half that

so your error is approximately $\frac{2}{10,000,000}=\frac{1}{5,000,000}$.

If you sum $n$ terms, the error is not exactly $\frac1n$; a better expression is $$\pi\approx 4 \sum_{k=1}^{n} \frac{(-1)^{k+1}}{2 k-1} +(-1)^n\left(\frac1n- \frac{1}{4n^3}+ \frac{5}{16n^5}- \frac{61}{64n^7}\right)+O\left(\frac{1}{n^9}\right)$$

and for example when $n=5$ the partial sum is wrong in the first decimal place, but with the adjustment the approximation is correct up to five decimal places. The numerators in the adjustment are in OEIS A000364 or signed in A028296.

Coming to $n=50000000$, the powers of $5$ largely soak up the powers of $2$ in the denominator so the error terms become successively smaller and seem separate: $+0.0000002$ $ -0.000000000000000000002$ $ + 0.0000000000000000000000000000000001$ $ -0.0000000000000000000000000000000000000000000000122$ $ +\cdots$ and so only a few of the decimal places are changed, as you noticed in your calculations

3.14159245358979323846464338327950278419716939938730582097494182230781640
3.14159265358979323846264338327950288419716939937510582097494459230781640
        ^             ^            ^            ^^^          ^^^ 
        2            -2            1           -122          277  

Answer 2

The surprising thing--which is known to happen in this series but rarely happens with other series--is: many correct digits appear after the first incorrect digit. An interesting explanation is here:

Borwein, J. M.; Borwein, P. B.; Dilcher, K., Pi, Euler numbers, and asymptotic expansions, Am. Math. Mon. 96, No. 8, 681-687 (1989). ZBL0711.11009.