I know that is the sequence of partial sums of a series is convergent, then the series is convergent. But let's say that the sequence of partial sums converges to the value $2$. Does that mean the sum of the series is $2$?
That doesn't make much sense to me because: as an example consider the series whose partial sums are given by the formula $s_n = 2 - 3(0.8)^n$. The limit as $n$ goes to infinity of that function is $2$. But wouldn't the sum of the series be much higher? Since: $s_1 + s_2 + s_3 + ... = 2-3(0.8) + 2-3(0.64)+2-3(0.512)+...$
Two is added with each new partial sum, so how can the sum of the series be two?
Sorry if this is a stupid question, I'm likely misunderstanding something fundamental about series. Any help is appreciated.