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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}≤Cb_{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}) | ||

≤ | 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 proof--since the partial sums of the *b*_{n}
become arbitrarily large and

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