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](https://tex.z-dn.net/?f=x)
gets arbitrarily large, the function
![f(x)=\dfrac1x](https://tex.z-dn.net/?f=f%28x%29%3D%5Cdfrac1x)
will approach 0. You can see why this must be the case by checking what happens to the value of
![\dfrac1x](https://tex.z-dn.net/?f=%5Cdfrac1x)
when
![x](https://tex.z-dn.net/?f=x)
is picked to be 10, or 1000, or 1000000000, and so on. Clearly,
![\dfrac1x](https://tex.z-dn.net/?f=%5Cdfrac1x)
must be positive, but the large the denominator, the smaller the value of
![\dfrac1x](https://tex.z-dn.net/?f=%5Cdfrac1x)
. 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](https://tex.z-dn.net/?f=f%28x%29%3D%5Csin%20x)
will oscillate indefinitely between the values of -1 and 1, so this function is said to not converge to any specific value as
![x](https://tex.z-dn.net/?f=x)
increases indefinitely, which means
![\sin x](https://tex.z-dn.net/?f=%5Csin%20x)
diverges.