Interesting phenomenon with the $\zeta(3)$ series I noticed that if one takes certain partial sums of the series for $\zeta(3)$:
$$\zeta(3) = \sum_{n=1}^{\infty} \frac{1}{n^3} \approx \sum_{n=1}^{N} \frac{1}{n^3}$$
an interesting phenomenon occurs for some values of $N$. For example, with $N = 100000$, the sum is
$$1.2020569031095947853972381615115330296...$$
while the exact value is
$$1.2020569031595942853997381615114499908...$$
.
Note that there are stretches of agreement of the digits beyond the initial segment:
$$1.2020569031(0)9594(7)8539(72)38161511\ (\mathrm{pattern}\ \mathrm{ends})$$
where the parentheses represent disagreeing digits. Why does this happen, what values of "N" give the best "pseudo-approximations", and what is a proof of those answers?
 A: I was hoping someone else more knowledgeable would give a more detailed answer, but: the answer is the Euler-Maclaurin formula (applied to the difference between the two sums). The points at which the digits don't match correspond to the part of the formula involving a sum over Bernoulli numbers. The values of $N$ which most have this property are powers of $10$. 
A: The behaviour you discovered is very similar to the one exhibited by the Madhava–Leibniz sum for $\frac{\pi}{4}$:
$$
\frac{\pi}4 = \sum_{k=0}^\infty\frac{(-1)^k}{2k+1}
$$
The decimal expansion of $\pi$ obtained through this formula will contain digits that disagree with $\pi$ at digits predictable by a corresponding sum over Euler numbers. In particular, calculating the first five million digits of the decimal expansion will yield
$$
3.141592\underline{4}5358979323846\underline{4}643383279502\underline{7}841971693993\underline{873}058...
$$
where the underlined digits disagree with the decimal expansion of $\pi$.
The disagreeing digits can be found using
$$
\frac{\pi}{2} - 2 \sum_{k=1}^\frac{N}{2} \frac{(-1)^{k-1}}{2k-1} \sim \sum_{m=0}^\infty \frac{E_{2m}}{N^{2m+1}}
$$
In your case, these digits can can be found in a similar way using a sum over Bernoulli numbers.
