# How do I solve this infinite series? **URL:** \ **Category:** Factual Questions\ **Created:** [November 11, 2011, 9:53am UTC](https://boards.straightdope.com/t/how-do-i-solve-this-infinite-series/602583 "2011-11-11T09:53:43Z")\ **Posts on this page:** 4\ **Page:** 1
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [November 11, 2011, 9:53am UTC](https://boards.straightdope.com/t/how-do-i-solve-this-infinite-series/602583/1 "2011-11-11T09:53:43Z")
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.) ---
**Author:** ![Indistinguishable](https://avatars.discourse-cdn.com/v4/letter/i/90ced4/32.png) [@Indistinguishable](https://boards.straightdope.com/u/Indistinguishable)\ **Post date:** [November 11, 2011, 10:02am UTC](https://boards.straightdope.com/t/how-do-i-solve-this-infinite-series/602583/2 "2011-11-11T10:02:30Z")
> [@CookingWithGas](#): > > 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? 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. Thus, your post-tax income + your post-tax income \* r = your pre-tax income. In other words, your post-tax income \* (1 + r) = your pre-tax income. In other words, your post-tax income = your pre-tax income/(1 + r). In other words, the amount you pay in taxes = your post-tax income \* r = your pre-tax income/(1 + r) \* r. In other words, you are taxed at a rate of r/(1 + r) \* your pre-tax income. ---
**Author:** ![Wendell\_Wagner](https://avatars.discourse-cdn.com/v4/letter/w/8491ac/32.png) [@Wendell\_Wagner](https://boards.straightdope.com/u/Wendell_Wagner)\ **Post date:** [November 11, 2011, 10:12am UTC](https://boards.straightdope.com/t/how-do-i-solve-this-infinite-series/602583/3 "2011-11-11T10:12:44Z")
I’m guessing that the following is what you mean: You have received an amount of money X. What is the amount Y such that r% of Y, plus Y, equals X. The answer is that (1 + r%) times Y equals X. So say that you have received $300. You want to know what the amount X is such that 1 plus 50% equals $300. The answer is $200, since (1 + .50) times $200 equals $300. How do you than calculate that the answer is $200? You divide $300 by (1 + .50). $300 divided by 1.50 equals $200. Another example is that you have received $487. What is the number X such that X times (1 + 5%) equals $487? The answer to this is found by dividing $487 by 1.05, which equals approximately $463.81. ---
**Author:** ![CookingWithGas](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/cookingwithgas/32/485_2.png) [@CookingWithGas](https://boards.straightdope.com/u/CookingWithGas)\ **Post date:** [November 11, 2011, 10:28am UTC](https://boards.straightdope.com/t/how-do-i-solve-this-infinite-series/602583/4 "2011-11-11T10:28:00Z")
Thank you both. I made things way too complicated by thinking of this as an infinite series.