Matematické Fórum

Marian · 18. 12. 2013 13:41

Let

Prove that for every positive integer n the following estimate holds true:

Brano · 15. 06. 2014 01:25 — Editoval Brano (15. 06. 2014 01:37)

$|r_n|=\frac{1}{n+1}-\frac{1}{n+2}+\frac{1}{n+3}-\frac{1}{n+4}+...=\frac{1}{n+1}-\frac{1}{(n+2)(n+3)}-\frac{1}{(n+4)(n+5)}-...<$
$<\frac{1}{n+1}-\frac{1}{(n+2)(n+4)}-\frac{1}{(n+4)(n+6)}+...=\frac{1}{n+1}-\frac{1}{2}\left(\frac{1}{n+2}-\frac{1}{n+4}+\frac{1}{n+4}-\frac{1}{n+6}+...\right)=$
$=\frac{1}{n+1}-\frac{1}{2n+4}<\frac{1}{2n-1}$

Marian · 16. 06. 2014 06:48

Nice job.

My method works in a similar manner.

Brano · 16. 06. 2014 15:03

↑ Marian:
note that also holds
$\frac{1}{n+1}-\frac{1}{2n+4}<\frac{1}{2n}<\frac{1}{2n-1}$
which I think is somewhat nicer upper bound

Matematické Fórum

#1 18. 12. 2013 13:41

An estimate of the remainder of the anharmonic infinite series

#2 15. 06. 2014 01:25 — Editoval Brano (15. 06. 2014 01:37)

Re: An estimate of the remainder of the anharmonic infinite series

#3 16. 06. 2014 06:48

Re: An estimate of the remainder of the anharmonic infinite series

#4 16. 06. 2014 15:03

Re: An estimate of the remainder of the anharmonic infinite series

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