• Support PF! Buy your school textbooks, materials and every day products Here!

Show that a Sequence is monotonically decreasing

  • Thread starter Calu
  • Start date
  • #1
73
0
Member warned about deleting template parts

Homework Statement



I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n.

I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.
 

Answers and Replies

  • #2
Ray Vickson
Science Advisor
Homework Helper
Dearly Missed
10,706
1,728

Homework Statement



I was wondering how I would go about showing that (an) is monotone decreasing given that an = 1/√n.

I believe I have to show an ≥ an+1, but I'm not sure how to go about doing that.
Well, PF rules require you to make a start on your own. What do you know about the magnitudes of ##\sqrt{n}## and ##\sqrt{n+1}##?
 
  • #3
73
0
Well, PF rules require you to make a start on your own. What do you know about the magnitudes of ##\sqrt{n}## and ##\sqrt{n+1}##?
I know that the magnitude of ##\sqrt{n+1}## is larger than that of ##\sqrt{n}##. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.
 
  • #4
pasmith
Homework Helper
1,738
410
I know that the magnitude of ##\sqrt{n+1}## is larger than that of ##\sqrt{n}##. Therefore I would assume that the opposite would be true for the magnitude of their reciprocals which would make an≥an+1 as required, however I'm not sure how to write this in a more coherent way.
(1) [itex]\sqrt{n + 1} > \sqrt{n}[/itex].
(2) If [itex]n > 0[/itex] then dividing both sides by [itex]\sqrt{n} > 0[/itex] preserves the inequality. Hence [itex]\frac{\sqrt{n + 1} }{\sqrt{n}} > 1[/itex].
(3) Dividing both sides by ... > 0 preserves the inequality. Hence ...
 

Related Threads on Show that a Sequence is monotonically decreasing

  • Last Post
Replies
3
Views
2K
  • Last Post
Replies
8
Views
2K
  • Last Post
Replies
3
Views
2K
Replies
2
Views
995
  • Last Post
Replies
2
Views
823
  • Last Post
Replies
9
Views
13K
  • Last Post
Replies
8
Views
1K
Replies
9
Views
91K
Replies
2
Views
1K
Replies
10
Views
840
Top