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For sufficiently large value of , what does the above function closely approximate?
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The for-loop that adds a number to s in every iteration can be seen as a sum from 1 to y
i = 1 ∑ y something
In every iteration, the previous value of p is multiplied by i x , so the something that is added is a product of i terms for every i ≤ y .
i = 1 ∑ y j = 1 ∏ i j x = i = 1 ∑ y i ! x i
and these are exactly the first y terms of the Taylor Series of the exponential function e x , so for sufficiently large y , the sum converges to e x .