Explain the logic of solving the sum_series problem: how to construct the sum of two stepwise time series?

The sum_series task involves adding two stepwise time series, where values change discretely at specific moments in time. The logic of the solution is to:

1. Combine all unique time points from both series.
2. For each point, determine the current value of each series (the last known value before this point).
3. Sum these values to get the total at each time point.

This results in a new stepwise series that reflects the sum of the original series.

Example in Python using pandas:

import pandas as pd

# Two stepwise series
series1 = pd.Series([10, 20, 15], index=pd.to_datetime(['2023-01-01', '2023-01-03', '2023-01-05']))
series2 = pd.Series([5, 7, 3], index=pd.to_datetime(['2023-01-02', '2023-01-04', '2023-01-06']))

# Combine indices
all_times = series1.index.union(series2.index).sort_values()

# Recalculate values with forward fill
s1_filled = series1.reindex(all_times, method='ffill').fillna(0)
s2_filled = series2.reindex(all_times, method='ffill').fillna(0)

# Sum
sum_series = s1_filled + s2_filled
print(sum_series)

This will give the sum of the two series considering their stepwise nature.