Natural Blogarithms

Mathematical Explosions

November 15th, 2006 by SplineGuy

When teaching limits of sequences, I often use the phrase that some part of the general term “blows up”. I typically use it to indicate that it is going to infinity. For example, in the statement that $\displaystyle \frac{1}{n} \longrightarrow 0$ , I’d say that the denominator of the fraction blows up and causes the entire fraction to go to zero. I have heard it so often and use it so much that it just sounds natural. I realized that it is slightly unnatural to the less mathematically experienced. On a recent Calculus III Exam, a student was using the Alternating Series Test for Convergence on the expression $\displaystyle \sum_{n=1}^{\infty} \frac{(-1)^n n}{2^n}$ , and he stated, “The denominator is going to explode. . .” He’s right, it does.