Math Is Fun Forum

mrpace

Member

Registered: 2012-08-16

Posts: 88

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?

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: Does this sequence converge or diverge?

What is the difference between oscillating and periodic?

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'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
I'm not crazy, my mother had me tested.

Bob

Administrator

Registered: 2010-06-20

Posts: 10,643

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

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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
Sometimes I deliberately make mistakes, just to test you!  …………….Bob

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Does this sequence converge or diverge?

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

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“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.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

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

I thought it was the other way around.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: Does this sequence converge or diverge?

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

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“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.