# How do I solve this infinite series?

I can’t remember how to do this but I think maybe I need to solve an integral. I know how to solve it by iterating until it converges within a given error but I am wondering if it can be solved exactly with an equation.

I am going to describe this as a hypothetical tax problem. I do not think there is any tax like in this universe but bear with me as this is an intellectual exercise.

I have a certain amount of income x. I must pay taxes on my after-tax income at a rate of r. That is, the rate is applied to my income after the tax is subtracted. But that means I need to know the amount of tax before I can calculate it, sort of a “this sentence is false” tax paradox. Is there an equation for that?

(This same kind of calculation may actually apply in the case where an employee receives a company benefit that is taxable, and the company does not want the employee to have to bear the tax burden so does a “gross up” of the benefit so that it will equal some number net taxes. When I started my job, my new employer paid me to cover the educational expenses that my prior employer required me to reimburse upon leaving. That was paid as a taxable bonus, so they increased it to cover the taxes.)

Your post-tax income + the amount you pay in taxes = your pre-tax income.

Also, your post-tax income * r = the amount you pay in taxes.