Question

What is the difference between convergence and divergence? I'm in advanced algebra so try not to make a complicated answer. Thank you.

Answer 1

The concept of convergence is a topic more suited to a discussion of calculus if you want a fairly rigorous explanation.

But what it basically means is that, given a function that depends on one or more independent variables, the value of the function will "converge" or approach a finite value as the independent variable(s) approaches their own finite values.

Divergence means the opposite. If there is no finite or fixed value that a function appears to be approaching, then it does not converge, and is thus said to diverge.

Some examples: As $x$ gets arbitrarily large, the function

$f(x)=\dfrac1x$

will approach 0. You can see why this must be the case by checking what happens to the value of $\dfrac1x$ when $x$ is picked to be 10, or 1000, or 1000000000, and so on. Clearly, $\dfrac1x$ must be positive, but the large the denominator, the smaller the value of $\dfrac1x$. It will never actually take on the value of 0, but we can see that it must *converge to* 0.

On the other hand, the function

$f(x)=\sin x$

will oscillate indefinitely between the values of -1 and 1, so this function is said to not converge to any specific value as $x$ increases indefinitely, which means $\sin x$ diverges.