Suppose (a_{n}) is a monotonically decreasing sequence of positive numbers, and that converges.

Show that

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- Mar 17th 2013, 07:37 AMKanwar245Monotonically Decreasing Sequence
Suppose (a

_{n}) is a monotonically decreasing sequence of positive numbers, and that converges.

Show that - Mar 17th 2013, 10:01 AMMINOANMANRe: Monotonically Decreasing Sequence
This is a well known theorem in the series of positive terms.

If the series Σan of positive monotone decreasing terms is to converge then we must have not only lim(an)=0 but also lim(n(an))=0 . However the condition lim(n(an))=0 is only necessary , not a sufficient one for these type of series. If lim(nan) does not tend to zero then definitely the series diverges but lim(nan)=0 does not necessarily implies anything as to the possible convergence of the series. in fact the Abel series Σ(1/nlogn) diverges though lim(nan)=0

Get a good book on series to revise all these theorems.

MINOAS - Mar 18th 2013, 04:39 PMKanwar245Re: Monotonically Decreasing Sequence
- Mar 18th 2013, 08:18 PMhollywoodRe: Monotonically Decreasing Sequence
No, that definitely doesn't work.

But you know (by definition) that . So you just need to show that the finite sum is less than .

- Hollywood