Cesàro summation

Cesàro summation is a way of finding the "average" of a series of numbers when the series itself might not have a defined sum.

Imagine you have a bunch of toy cars, and you want to find out how many you have. You start counting them one by one, but you quickly get lost and can't remember which ones you already counted. So instead, you start grouping them together in sets of 3. You count the first 3 cars, then the next 3, and so on until you've counted all of them.

Now, instead of adding up all the cars in the series, you can find the average number of cars per group by adding up the number of cars in each group and dividing by the total number of groups. This is what Cesàro summation does for a series of numbers.

For example, let's say we have the series 1, -1, 1, -1, 1, -1.... This series keeps going on forever, switching between 1 and -1. It doesn't have a defined sum because it keeps alternating between positive and negative values.

To find the Cesàro sum of this series, we group the numbers together in sets of increasing size:
- Group 1: 1
- Group 2: 1, -1
- Group 3: 1, -1, 1
- Group 4: 1, -1, 1, -1
- Group 5: 1, -1, 1, -1, 1
And so on...

Then, we find the average of each of these groups:
- Group 1: 1/1 = 1
- Group 2: (1-1)/2 = 0
- Group 3: (1-1+1)/3 = 1/3
- Group 4: (1-1+1-1)/4 = 0
- Group 5: (1-1+1-1+1)/5 = 1/5
And so on...

Finally, we add up all the averages and divide by the total number of groups we've calculated so far:
(1+0+1/3+0+1/5+...) / (1+2+3+4+5+...)

This gives us the Cesàro sum of the series, which is 1/2. This means that if we "average out" the series by grouping the numbers, the result approaches 1/2 as we group more and more numbers together.

In summary, Cesàro summation is a way of finding the "average" of an infinite series of numbers that might not have a defined sum, by grouping the numbers together and calculating the average of each group.