Getting Friendly
It turns out that the sequence of terms and the sequence of partial sums of a series are not distant second cousins. They're both born from the same mother series, so they're siblings. Sometimes they look just alike, similar to identical twins. Sometimes they look as different as peas and carrots. They go together, nonetheless.
Look at any series
.
We can make a sequence out of the terms of the series:
a1, a2, a3, ...
We can also make a sequence out of the partial sums of the series:
S1, S2, S3, ....
These two sequences are different. They won't necessarily converge to the same value. They might not both converge. However, they are related in one key way.
If the sequence of partial sums converges, then the sequence of terms must converge to zero.
We know that saying, "the sequence of partial sums converges," is the same as saying, "the series converges." Here's a another mind-blowing fact.
If a series converges, then its terms must converge to zero.
Mathematician and magician are two very different professions. Because we don't like waving our hands, turning around three times and spitting to prove things, we're going to show both of these are true using math.