Best IRS letter ever

[SwissTony]

4. Skip to ***HERE*** if you're not interested in the maths

The infinite series (the last component) can be evaluated as

S = 1/2 + 1/(2^2) + 1/(2^3) + ...

where S is the label form the sum and "^" is the exponentiation operator (2^3 = 2 to the third power = 2*2*2 = 8)

A simpler way to write it avoiding algebraic notation is

S = 1/2 + 1/4 + 1/8 + 1/16 +...

If we temporarily ignore the first term (1/2), we can extract 1/2 from ALL the remaining terms

S = 1/2 *(1/2)*(1/2 + 1/4 + 1/8 + ...)

Now if you look closely, the last term in brackets is just what I called S, so the equation now reads

S = 1/2 + (1/2)*S

Subtract (1/2)S from both sides gives

(1/2) S = 1/2

i.e. S=1

The second term is more complex. It comes from something called de Moivre's theorem, one of the results of which is that

e^(pi*i) = -1

pi is the old familiar 3.14, ratio of circumference to diameter of a circle

e is a good friend of pi and pops up lots in maths. It's roughly equal to 2.718

So that's the value of the last two terms, -1 and +1 respectively.



***HERE***


So the expression evaluates to

T = 0.002 -1 +1 = 0.002

i.e. 2 tenths of one cent. Mr Munroe is justifiably peeved at receiving this bill.