Infinite Geometric Series

An infinite geometric series adds terms that keep getting multiplied by the same factor r each time: a1, a1r, a1r^2, and so on. When the absolute value of r is less than 1, the terms shrink toward 0, so the total sum settles to a fixed value rather than growing without bound. The formula comes from comparing the series to a shifted copy of itself and using the fact that the leftover tail vanishes as the number of terms becomes very large.

Use this when a quantity is repeated over and over by a constant ratio and the repeated pieces get smaller, such as in some finance, probability, decay, or limit problems. It only applies when the common ratio satisfies |r| < 1, so the terms actually converge.

• Using the formula when |r| is 1 or greater, which makes the infinite sum diverge
• Forgetting that a1 is the first term, not the ratio or the second term
• Substituting the wrong sign for r, especially when the terms alternate