sequences and series - Formula for partial sum of (10)/(10n+1)

I'm trying to find $S_n$ of an infinite series, and I'm having trouble. Here is the equation:

$$\sum_{n=1}^\infty \frac{10}{10n+1}$$

This gives me these terms:

$$S_n = \frac{10}{11}+\frac{10}{21}+\frac{10}{31}+\frac{10}{41}+ ... + \frac{10}{10n+1}$$

After I calculate the terms of $S_n$, I get:

$$S_n = \frac{10}{11},\frac{20}{32},\frac{30}{63},\frac{40}{104}, ... $$

Obviously the top is 10n, but I'm having trouble with the bottom. I recognize a pattern in the differences of the terms, mainly that each is separated by the previous difference + 10:

$$S_2-S_1=21$$

$$S_3-S_2=31$$

$$S_4-S_3=41$$

But I have no idea how to translate that into a formula. Note that I am aware that the series diverges, but I would still like to create a formula with which I can take the limit of infinity to verify that it diverges. Any suggestions?

EDIT: Apparently I'm asking the wrong question. What I'm essentially trying to figure out is how to determine whether the series converges or diverges based on the information available. I can use intuition to come to the conclusion it's divergent, but how do I do it mathematically?