Expanding Average does not return the same result in pandas and polars

[eitsupi]

The two results in the following code block appear to be different.

modern-polars/book/timeseries.qmd

Lines 152 to 175 in 8565ca3

	``` {python}
	close = pl.col("close")
	ohlcv_pl.select(
	[
	pl.col("time"),
	close.alias("Raw"),
	close.rolling_mean(28).alias("28D MA"),
	close.alias("Expanding Average").cum_sum() / (close.cum_count() + 1),
	close.ewm_mean(alpha=0.03).alias("EWMA($\\alpha=.03$)"),
	]
	).to_pandas().set_index("time").plot()
	plt.ylabel("Close ($)")
	```
	## Pandas
	``` {python}
	ohlcv_pd["close"].plot(label="Raw")
	ohlcv_pd["close"].rolling(28).mean().plot(label="28D MA")
	ohlcv_pd["close"].expanding().mean().plot(label="Expanding Average")
	ohlcv_pd["close"].ewm(alpha=0.03).mean().plot(label="EWMA($\\alpha=.03$)")
	plt.legend(bbox_to_anchor=(0.63, 0.27))
	plt.ylabel("Close ($)")
	```