Write the first few terms:. The sequenceis nondecreasing.To determineconver-
gence,we see if we can find a constantBsuchthat. If we cannotfind sucha constant,then
the sequencediverges.
If two fractionshavethe samenumeratorbut differentdenominators,the fractionwith the smallerdenomi-
natoris the largerfraction.Thus,. Then and, in fact,
.
SeriesWithoutNegativeTerms(harmonic,geometric,p-series)
Thereare severalspecialkindsof serieswith nonnegativeterms,i.e., termsthat are eitherpositiveor zero.
We will studythe convergenceof suchseriesby studyingtheir correspondingsequencesof partialsums.
Let’s start with theharmonicseries:'
..
The sequenceof partialsumslook like this:
In orderfor the harmonicseriesto converge,the sequenceof partialsumsmustconverge.The sequence
of partialsumsof the harmonicseriesis a nondecreasingsequence.By the previoustheorem,if we find a
boundon the sequenceof partialsums,we can showthat the sequenceof partialsumsconvergesand,
consequently, that the harmonicseriesconverges.
It turnsout that the sequenceof partialsumscannotbe madeless than a set constantB. We will omit the
proofhere,but the mainidea is to showthat the a selectedinfinitesubsetof termsof the sequenceof partial
sumsare greaterthan a sequencethat diverges,whichimpliesthat the sequenceof partialsumsdiverge.
Hence,the harmonicseriesis not convergent.
We can also workwithgeometricserieswhosetermsare all non-negative.