Advanced High-School Mathematics

266 CHAPTER 5 Series and Differential Equations

then the sequenceconvergesto some limitL(which we might not be
able to compute!): limn→∞an = L.

 -

6

?

M

r

r r

r r r r r

(n,an)

r r

Figure 1

So what dosequenceshave to do withinfinite series? Well, this

is simple: if each terman in the infinite series

∑∞

n=0

an is non-negative,

then thesequence of partial sumssatisfies

a 0 ≤ a 0 +a 1 ≤ a 0 +a 1 +a 2 ≤ ··· ≤

∑k

n=0

an≤···.

Furthermore, if we can establish that for some M each partial sum

Sk =

∑k

n=0

ansatisfiesSk≤M then we have a limit, say, lim

k→∞

Sk=L, in

which case we write

∑∞

n=0

an = L.

In order to test a given infinite series

∑∞

n=0

anof non-negative terms for

convergence, we need to keep in mind the following three basic facts.

Fact 1:In order for

∑∞

n=0

anto convergeit must happen that nlim→∞an=

0. (Think about this: if the individual terms of the series don’t get
   small, there’s no hope that the series can converge. Furthermore,
   this fact remains true even when not all of the terms of the series
   are non-negative.)