What does the false infinite sum of a series mean? For any geometric series with |$r$| &lt; 1 , I know that

$$\sum_{k=1}^{∞} ar^{k-1} =\frac{a}{1-r}$$

But if |$r$| > 1 and you try to use the formula, you'll get a weird answer. For instance:

$$4+8+16+32+64+128+... =\sum_{k=1}^{∞} (4)2^{k-1}= \frac{4}{1-2} = -4$$ 

That answer obviously doesn't make sense; the series diverges. So what does -4 mean? Where did it come from, and how is it related to the series? It must be significant somehow.
For a truncated series, we have

$$\sum_{k=1}^Kar^{k-1}=a\frac{1-r^K}{1-r} \tag 1$$

For $a=4$ and $r=2$, use of $(1)$ reveals 

$$\sum_{k=1}^K4\left(2\right)^{k-1}=4\frac{1-2^K}{1-2}=2^{K+2}-4$$

The $-4$ appears as the "convergent component" of the divergent series (as $K\to \infty$).
This is a way
of assigning meaning
to an otherwise divergent sum.

Look up "divergent series."

The subject is large.