fx-991CW: Sums with a Step Not Equal to One
Step ≠ 1? An Adjustment is Necessary
The summation function on the Casio fx-991 CW calculator, and a lot of other advanced scientific calculators, calculates the sum of f(x) from x = a to x = b, with an increment of 1.
But what if have a sum of following:
log(2) + log(4) + log(6) + log(8): sum(f(x) from x = 2 to x = 8, step = 2)
Clearly, the increment is not 1, but 2 instead. Believe it or not, we can still use the summation function. We will need to adjust both the function and the limits to fit the requirements of the summation function.
1. Adjust f(x) to f(step * x)
2. Adjust the limits to x = a ÷ step and x = b ÷ step
For this particular problem:
1. f(x) becomes log(2 * x)
2. The limits become x = 2 ÷ 2 = 1 to x = 8 ÷ 2 = 4 (this allows the step to be 1)
Which transforms:
sum(log(x) from x = 2 to x = 8, step = 2)
to:
Σ(log(2 * x) from x = 1 to x = 4)
Sum: 2.584331224
A Couple More Examples
Example 1:
sum(x^2 from x = 0.5 to x = 2, step = 0.5), f(x) = x^2
The key here is the step (also known as the increment). The adjustments needed are:
1. f(x) becomes (0.5 * x)^2 = (x / 2)^2
2. The limits become x = 0.5/0.5 = 1 to x = 2/0.5 = 4
Sum: 15/2 = 7.5
Example 2:
sum(e^(x) from x = 1.5 to x = 2.1, step = 0.1), f(x) = e^(x)
The adjustments needed are:
1. f(x) becomes e^(0.1 * x) = e^(x / 10)
2. The limits become x = 1.5/0.1 = 15 to x = 2.1/0.1 = 21
Sum: 43.1994368
In Summary…
For the sum:
sum(f(x), x = a to x = b, step = Δx):
1. Adjust f(x) to f(Δx * x)
2. Adjust the limits: x = a/Δx to x = b/Δx
3. Summation function looks like this: Σ( f(Δx * x), x = a/Δx to x = b/Δx)
Hope you find this useful! Take care,
Eddie
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