Every infinite series can be thought of as a an infinite sequence. The second column is ak, and the third is Sk. Note that the ak's are approaching 0, while the Sk appear to approach some son-zero limit. To help make the difference between a sequence and its corresponding series more apparent , lets look at both graphically. Each when a series diverges to infinity, the journey may be quite leisurely.
Lets examine the harmonic series, which is the sum of the reciprocals of the natural numbers.
First we'll add up the first 100 reciprocals and see how close we are to infinity. Let add up the first 1,000 terms and see what progress is made.