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Homework Statement
Let ##\sum a_n## be a convergent series of non-negative terms and suppose that the sequence ##\{a_n\}## is non-increasing.
Apply the Cauchy Criterion to show that ##\forall ε > 0##, if m and n are sufficiently large, then ##na_n < (ε/2) + ma_n##.
Hint : ##a_{m+1} + ... + a_n ≥ (n-m)a_n## for ##n>m##.
Use this fact ( fix m ) to show that ##na_n → 0## as ##n → ∞##.
Give an example of a series where ##na_n → 0## as ##n → ∞##, ##\{a_n\}## is not increasing and yet ##\sum a_n## diverges.
Homework Equations
Cauchy Criterion :
A sequence ##\{a_n\}## converges ##⇔ \forall ε > 0, \exists N \space | \space n,m > N \Rightarrow |a_n - a_m| < ε##.
The Attempt at a Solution
First time I've seen something like this. I'm not sure what the question is asking me to do in particular. Then again I'm not quite sure where to start this one?
So if ##na_n < (ε/2) + ma_n##, then ##(n-m)a_n < ε/2##. I have a feeling the hint comes into play somehow here, but I'm not seeing how it all comes together here. ( I'm thinking I should assume W.L.O.G, that n>m ).
Also as for the example portion of the question, ##\{a_n\} = \{ \frac{1}{n^{3/2}} \}## works just fine I can see that.
If anyone could help me out a bit it would be much appreciated.
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