Know which types of equations have infinity as their answer. You must first know that most infinite problems are series. More specifically, they are divergent series. Calculus defines this type of series as an infinite sequence of numbers whose sum’s limit approaches infinity. In other words, the answer to this type of equation is too large to comprehend.
Consider the methods of solving this type of problem. There are several ways of going about it. According to notes taken from an advanced calculus class, you can solve it eight ways: Nth term test, geometric series test, p-series test, direct comparison test, limit comparison test, ratio test and integral test. For extra practice on this, click the \"Old Dominion University Infinite Series Tutorial\" link under Resources.
Choose a method to solve. This step depends on what the original equation is. For example, a p-series test is used for an equation that follows this model: ?(n=1,?) 1/n^p. If \"p\" is greater than 1, then the series is infinite. Consult your calculus textbook for other examples of testing an infinite series.
Set up the formula. For the sake of discussion, we will solve an equation together using the ratio test. Our sample equation is:
?(n = 1,?) n!e^-n
Following algebra rules, you can write \"n!e^-n\" as \"n!/e^n.\" The ratio test formula is lim (n??): An + 1/An.
Substitute variables. Plug “n + 1” where “n” is in the original equation and divide that by the original equation. When simplifying the formula, it should look like this:
lim (n??): [n!(n + 1) / e^(n + 1)] * (e^n) / n!
Simplify the equation. The first part of solving is canceling what is no longer needed. Assuming you know how to cross-cancel from algebra, “n!” on top and “n!” below cancels to become 1. “e^n” on top and “e^(n+1)” below becomes “e.” You are left with:
lim (n??): (n + 1) / e
Solve the equation using the \"lim (n??)\" model. Plug “?” for n. “? + 1” equals infinity (?). Because “e” is a finite number, infinity divided by \"e\" is infinity. So according to the ratio test, this solution diverges to infinity.