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Until the final section of this chapter, we will restrict our attention to series with an≥ 0. Thus the partial sums are increasing:

s1≤s2≤ ... ≤sn≤ ...    

If the series an is to converge, there must be some B such that sn≤B for all n, or the sn 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 {sn} 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 an, bn, with an, bn≥ 0 for all n, suppose there exists a number C > 0 such that

an≤Cbn    

for all n and that bn converges. Then an converges and

an≤Cbn    

To prove this statement, it suffices to show that the number Cbn is a bound for the partial sums a1 + a2 + ... + an. Then the least upper bound of these partial sums must exist and is clearly less than or equal to Cbn. Thus we need only note that


a1 + ... + an≤Cb1 + ... + Cbn  
 =C(b1 + ... + bn)  
 ≤Cbn  

A similar test enables us to show that certain series diverge. If an and bn are again two series with an, bn≥ 0 for all n, suppose that there exists C≥ 0 such that an≥Cbn for all n and that bn diverges. Then an also diverges. The proof of this fact is similar to the previous proof--since the partial sums of the bn become arbitrarily large and

a1 + ... + an≥Cb1 + ... + Cbn = C(b1 + ... + bn)    

the partial sums of the an also become arbitrarily large.

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