What is this formula??

Question

I am in class for computer science and have to write a program that utilizes the following "formula" (which I have come to find out this instructor used software to hand write):

I can't find it anywhere, no matter where I look, and I can't find anything similar. Please help.

Her stupid, cryptic instructions never give any insight into anything, and I am at my wit's end.

Answer 1

As a computer programmer, I do not need to know what the formula is for. I need to recognize its pattern and be able to write code that can calculate the formula correctly for a given n. Here is one possible approach:

If n is 1 (or less than 1), then y is 1.

Otherwise:

Set y to 1.

For each num from 2 to n, inclusive:

If num is even:

Add 3^num / num! to y

Else:

Subtract 3^num / num! from y

Answer 2

Juliet C's answer is correct, both the program (for the formula *as given*!) and the remark that you don't need to know anything about the formula in order to program it (but maybe you *should* know something about it ... see "NOTE" further down.)

Anyway, you might still be curious about what this is the formula *for*.

Except for one error, it is the formula for e^(-3), that is, the constant e=2.71828... raised to the power -3, which is the same as 1/(e^3), that is, 1 divided by e cubed.

You can do it on a calculator using the e^x ("e to the x") button. That would show up on the calculator screen as "e(-3)", where "-" is the "negative" key, not the "subtract" key.

(* Note) The program is correct for the formulas *as given*! But the formula as given is incorrect! The error in the formula is that it should be "1 - 3 + all the fractions" -- the -3 got left out.

It can also be written as 3^0/0! - 3^1/1! + 3^2/2! - 3^3/3! + ... (-1)^n/n!

So the program for the *correct* formula would be

Set y to 0

For each num from 0 to n, inclusive:

If num is even:

Add 3^num / num! to y

Else:

Subtract 3^num / num! from y

The formula can also be written as (-3)^0/0! + (-3)^1/1! + (-3)^2/2! + (-3)^3/3! + ... +(-3)^n/n!

So the program becomes simply

Set y to 0

For each num from 0 to n, inclusive:

Add (-3)^num / num! to y

NOTE: If you were working for an employer who asked you to program the formula *as given* ... they might complain that your program has a bug because it gave the wrong answer! And you would have to say "I programmed exactly what you asked me to program," and neither one of you would know why the answer was wrong.

So that's why it's good as a programmer to know some other things besides just programming. That way you could say to your employer "Did you mean for me to write a program to calculate e^(-3)? That should be a just slightly different formula."

(You should NOT just go ahead and change the formula without asking the employer, because maybe they DID want exactly that formula, and DID NOT want e^(-3) ... although I doubt it.)

And if your employer DOES want e^(-3) and you are using a language which has a function to calculate powers (usually called something like "exp"), you could get an even more exact answer just by saying

PRINT exp(-3)