Until the final section of this chapter, we will restrict our attention to series with a_{n}≥ 0. Thus the partial sums are increasing:
s_{1}≤s_{2}≤^{ ... }≤s_{n}≤^{ ... } 

If the series a_{n} is to converge, there must be some B such that
s_{n}≤B for all n, or the s_{n} will become arbitrarily large. Such a B is called
an upper bound. The value to which the series converges is the least of all possible
upper bounds. It turns out that whenever the sequence {s_{n}} of partial sums has an
upper bound, there exists a least upper bound, to which the series converges. This fact
enables us to prove the comparison test, stated below.
For two series a_{n}, b_{n}, with a_{n}, b_{n}≥ 0 for all n, suppose there exists a number C > 0 such that
for all n and that b_{n} converges. Then a_{n}
converges and
a_{n}≤Cb_{n} 

To prove this statement, it suffices to show that the number Cb_{n} is
a bound for the partial sums a_{1} + a_{2} + ^{ ... } + a_{n}. Then the least upper bound of these
partial sums must exist and is clearly less than or equal to Cb_{n}.
Thus we need only note that
a_{1} + ^{ ... } + a_{n}  ≤  Cb_{1} + ^{ ... } + Cb_{n} 

 =  C(b_{1} + ^{ ... } + b_{n}) 

 ≤  Cb_{n} 

A similar test enables us to show that certain series diverge. If a_{n}
and b_{n} are again two series with a_{n}, b_{n}≥ 0 for all n,
suppose that there exists C≥ 0 such that a_{n}≥Cb_{n} for all n and that
b_{n} diverges. Then a_{n} also diverges. The
proof of this fact is similar to the previous proofsince the partial sums of the b_{n}
become arbitrarily large and
a_{1} + ^{ ... } + a_{n}≥Cb_{1} + ^{ ... } + Cb_{n} = C(b_{1} + ^{ ... } + b_{n}) 

the partial sums of the a_{n} also become arbitrarily large.