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#1 2014-07-29 17:09:31

mrpace
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Registered: 2012-08-16
Posts: 63

Does this sequence converge or diverge?

n+1
(-1 )^(n-1) * 2n+1

what happens as n goes to infinity?

My thoughts are that it oscillates between near 0.5 and -0.5 but never hits either. does this mean it diverges?

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#2 2014-07-29 20:23:57

bob bundy
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Registered: 2010-06-20
Posts: 7,182

Re: Does this sequence converge or diverge?

Did you mean

When I first learnt about convergence, the implication was that if it didn't converge then it diverged.  But, more recently, I have met a text that listed four possibilities:

converge, diverge, oscillate and periodic.

Putting a name to it, is not as important as describing what the sequence is doing.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#3 2014-07-30 00:34:03

Agnishom
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From: Riemann Sphere
Registered: 2011-01-29
Posts: 21,923
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Re: Does this sequence converge or diverge?

What is the difference between oscillating and periodic?


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#4 2014-07-30 01:49:32

bob bundy
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Registered: 2010-06-20
Posts: 7,182

Re: Does this sequence converge or diverge?

oscillating: eg. 1, -1, 1, -1 ,1, -1, .............

periodic: eg.  1,2,3,5,1,2,3,5,1,2,3,5, ..........

I suppose you could argue that oscillating sequences are a special case of periodic.

Bob


Children are not defined by school ...........The Fonz
You cannot teach a man anything;  you can only help him find it within himself..........Galileo Galilei

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#5 2014-07-30 03:03:50

anonimnystefy
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From: Harlan's World
Registered: 2011-05-23
Posts: 15,937

Re: Does this sequence converge or diverge?

sin(n) would be oscillating, but not periodic.


“Here lies the reader who will never open this book. He is forever dead.
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#6 2014-07-30 06:22:18

ShivamS
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Registered: 2011-02-07
Posts: 3,646

Re: Does this sequence converge or diverge?

bob bundy wrote:

oscillating: eg. 1, -1, 1, -1 ,1, -1, .............

periodic: eg.  1,2,3,5,1,2,3,5,1,2,3,5, ..........

I suppose you could argue that oscillating sequences are a special case of periodic.

Bob

I thought it was the other way around.

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#7 2014-07-30 10:00:45

anonimnystefy
Real Member
From: Harlan's World
Registered: 2011-05-23
Posts: 15,937

Re: Does this sequence converge or diverge?

That sounds right. The only exception (that I can think of) is any constant sequence.


“Here lies the reader who will never open this book. He is forever dead.
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment
The knowledge of some things as a function of age is a delta function.

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