What makes a function diverges

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Upcoming Events. Featured on Meta. Now live: A fully responsive profile. The unofficial elections nomination post. Linked 0. Related Hot Network Questions. One reason for providing formal definitions of both convergence and divergence is that in mathematics we frequently co-opt words from natural languages like English and imbue them with mathematical meaning that is only tangentially related to the original English definition.

When we take two such words which happen to be opposites in English and give them mathematical meanings which are not opposites it can be very confusing, especially at first.

These may seem like opposites at first but they are not. Mathematicians have learned to be extremely careful about this sort of thing. In the case of convergence and divergence of a series, even though these words are actually opposites mathematically every sequence either converges or diverges and no sequence converges and diverges it is better to say this explicitly so there can be no confusion.

However there are several ways a sequence might diverge. This is an illustration of how in mathematics sometimes you have to use your imagination or guess and see what happens. Don't get anxious--instead, relax, take a deep breath and explore. For example, alternatively we could have done the following:. Does it converge or diverge? For large values of , the term very quickly goes to 0 , so we expect this to diverge, since diverges.

For , we have , so for all we have. In many cases looking at graphs helps a lot. However, on one side of the asymptote the graph approaches negative infinity, and on the other side it approaches positive infinity. We cannot determine what the true limit is, so we say that it does not exist saying that it diverges is also acceptable. Here is another example that is very common. It relies on something known as asymptotes , lines with special slopes that the graph of a function mimics.

Generally a graph will approach horizontal asymptotes only as the argument approaches positive or negative infinity. A vertical asymptote is a vertical line with an undefined slope.