According to the video https://www.youtube.com/watch?v=jcKRGpMiVTw, the series $s_n=\sum_{k=1}^n (-1)^{k+1} k$ is $2$-Cesàro summable. Because of the not-quite rigorous identity
$$ s_\infty = \frac{1}{(1+1)^2},$$
I expected that $s_n$ would then be $2$-Cesàro summable to $1/4$. However, computing the partial averages of the partial averages of $s_{n\rightarrow \infty}$ with the following Python code yields a sequence that quickly decays past 1/4 and seems to be tending towards 0.
def secondNeg(n):
if n % 2:
yield n
else:
yield -n
yield from secondNeg(n+1)
def Cesaro(mean, pos, s):
result = (mean*pos+next(s))/(pos+1)
yield result
yield from Cesaro(result, pos+1, s)
for i in Cesaro(0, 1, Cesaro(0, 1, secondNeg(1))):
print(i)
As a matter of fact, I obtained that the 985th average of averages is approximately $0.0015618150237966225$.
How do I prove that $s_n$ is (or isn’t) $2$-Cesàro summable, and how do I rigorously compute the value of this sum? And why does the code above not give a sequence converging to $1/4$?