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
a _{n}≤Cb _{n} |
for all n and that b _{n} converges. Then a _{n} converges and
a _{n}≤C b _{n} |
To prove this statement, it suffices to show that the number C b _{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 C b _{n} . Thus we need only note that
a _{1} + ^{ ... } + a _{n} | ≤ | Cb _{1} + ^{ ... } + Cb _{n} | |
= | C(b _{1} + ^{ ... } + b _{n}) | ||
≤ | C b _{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 proof--since 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.