# Does a geometric series oscillate?

I’ve been asked to in my own words, explain how to determine if a series converges, diverges, and oscillates.

Pretty straightforward but I was under the impression when we talk about oscillating, this is how we'd describe a sequence that alternates between - and + or a periodic sequence because they follow a specific pattern.

Sure, I could say the geo sequence 2, -1, 0.5, -0.75, 0.125 as S₄=2+(-1)+ 0.5 + (-0.75) oscillates, but I've not read anything that refers to a series that oscillates.
Original post by KingRich
I’ve been asked to in my own words, explain how to determine if a series converges, diverges, and oscillates.
Pretty straightforward but I was under the impression when we talk about oscillating, this is how we'd describe a sequence that alternates between - and + or a periodic sequence because they follow a specific pattern.
Sure, I could say the geo sequence 2, -1, 0.5, -0.75, 0.125 as S₄=2+(-1)+ 0.5 + (-0.75) oscillates, but I've not read anything that refers to a series that oscillates.

If you think about the sequence, then the sum to n (series) will follow a similar pattern. Im not sure exactly how theyre defining the terms but you could have |r|>1 then diverging, |r|<1 then converging, r=-1 then oscillating. Though the oscillating could refer to r<0.

If youre unsure, just pick some simple sequences with the above properties and write down the first few terms of the corresponding series (sum to n) and, as you say, put it in your own words with a suitable explanation.
(edited 3 weeks ago)
Original post by mqb2766
If you think about the sequence, then the sum to n (series) will follow a similar pattern. Im not sure exactly how theyre defining the terms but you could have |r|>1 then diverging, |r|<1 then converging, r=-1 then oscillating. Though the oscillating could refer to r<0.
If youre unsure, just pick some simple sequences with the above properties and write down the first few terms of the corresponding series (sum to n) and, as you say, put it in your own words with a suitable explanation.

At the moment, I am a little torn because there’s conflicting answers.

Consider the oscillating geometric sequence 1,-1,1,-1,1 a=1 and r=-1.. the accompanying oscillating geometric series will be 1-1+1-1…

Let’s say I needed to find S₃, this would give me a value of 1, were as S₄=0. So, it oscillates in the term that the sum alternates between two sum values.

If we consider the sequence 2,-1,0.5,-1/4,1/8.. this is also oscillating sequence, ergo oscillating series but with r=-1/2… S₄=5/4 but S₁₀=1/512…

Also, I believe I read that an oscillating series can either diverge or converge depending on the r value, too
Original post by KingRich
At the moment, I am a little torn because there’s conflicting answers.
Consider the oscillating geometric sequence 1,-1,1,-1,1 a=1 and r=-1.. the accompanying oscillating geometric series will be 1-1+1-1…
Let’s say I needed to find S₃, this would give me a value of 1, were as S₄=0. So, it oscillates in the term that the sum alternates between two sum values.
If we consider the sequence 2,-1,0.5,-1/4,1/8.. this is also oscillating sequence, ergo oscillating series but with r=-1/2… S₄=5/4 but S₁₀=1/512…
Also, I believe I read that an oscillating series can either diverge or converge depending on the r value, too

Im guessing that the question is referring to the limiting behaviour, so what happens for large n, though you should have a definition somewhere if theyre asking a question about it. So oscillating is the 1,-1,1,-1, ... sequence you refer to and the corresponding series will oscillate between 1 and 0 (forever). If r=-1/2 say, then they probably say that converges as the oscillations decrease to zero.

That would be the usual interpretation.
Original post by mqb2766
Im guessing that the question is referring to the limiting behaviour, so what happens for large n, though you should have a definition somewhere if theyre asking a question about it. So oscillating is the 1,-1,1,-1, ... sequence you refer to and the corresponding series will oscillate between 1 and 0 (forever). If r=-1/2 say, then they probably say that converges as the oscillations decrease to zero.
That would be the usual interpretation.

Well that’s the issue. The only mention of convergence within the material is regarding an oscillating sequence. There was no mention of an oscillating series. It is something I’m having to find external sources for. It’s why I was a little confused in the beginning. I imagine it must be talking about the limiting value oscillating and how the choice of r changed whether It partially diverges between two numbers or converges while maintaining its oscillating pattern.
Original post by KingRich
Well that’s the issue. The only mention of convergence within the material is regarding an oscillating sequence. There was no mention of an oscillating series. It is something I’m having to find external sources for. It’s why I was a little confused in the beginning. I imagine it must be talking about the limiting value oscillating and how the choice of r changed whether It partially diverges between two numbers or converges while maintaining its oscillating pattern.

I think so. Youve really got the condition that it (infinite series) converges if |r|<1 and diverges if |r|>1 and if r=-1 it oscillates. Maybe do a quick google if necessary - there are a few explanations.
Original post by mqb2766
I think so. Youve really got the condition that it (infinite series) converges if |r|<1 and diverges if |r|>1 and if r=-1 it oscillates. Maybe do a quick google if necessary - there are a few explanations.

But, a series can still oscillate even if r=-1/2. From what I have concluded. For a geometric series to oscillate, it would mean the sequence must oscillate, too. This can only be achieved where r<0.

R must be negative in order to achieve the oscillated pattern within the sequence..

Therefore, when r=-1 we have the condition where the limit values oscillate between two values and then the other condition where r<0, condition r doesn’t equal -1, is that the limit will oscillate while converging as n approaches infinity….

Any sequence with r>0 will not oscillate because each term will be larger than the previous.
Original post by KingRich
But, a series can still oscillate even if r=-1/2. From what I have concluded. For a geometric series to oscillate, it would mean the sequence must oscillate, too. This can only be achieved where r<0.
R must be negative in order to achieve the oscillated pattern within the sequence..
Therefore, when r=-1 we have the condition where the limit values oscillate between two values and then the other condition where r<0, condition r doesn’t equal -1, is that the limit will oscillate while converging as n approaches infinity….
Any sequence with r>0 will not oscillate because each term will be larger than the previous.

Theyre (probably) talking about what happens as n->inf. Thats the usual interpretation, but if necessary, talk to the people who set the question.
Original post by KingRich
But, a series can still oscillate even if r=-1/2. From what I have concluded. For a geometric series to oscillate, it would mean the sequence must oscillate, too. This can only be achieved where r<0.
R must be negative in order to achieve the oscillated pattern within the sequence..
Therefore, when r=-1 we have the condition where the limit values oscillate between two values and then the other condition where r<0, condition r doesn’t equal -1, is that the limit will oscillate while converging as n approaches infinity….
Any sequence with r>0 will not oscillate because each term will be larger than the previous.

The mathematical definition of oscillation looks at the limiting behaviour at infinity. According to that definition, you would need $r \le -1$ for a series to have non-zero oscillation. (Whether you allow r < -1 and allow infinite divergent oscillation is a matter of taste, so to speak, although the wiki article indicates it's typically allowed).

https://en.wikipedia.org/wiki/Oscillation_(mathematics)#Oscillation_of_a_sequence

(I know it talks about sequence not series, but the behaviour of a series $(a_n)$ is always just the behaviour of the sequence $(b_n)$ defined by $b_n = \sum_{k=1}^n a_k$.)

Whether you want to trust that the people writing the course are expecting the correct mathematical definition (which requires around a year of university maths to understand) is another matter.
(edited 3 weeks ago)
Original post by DFranklin
The mathematical definition of oscillation looks at the limiting behaviour at infinity. According to that definition, you would need $r \le -1$ for a series to have non-zero oscillation. (Whether you allow r < -1 and allow infinite divergent oscillation is a matter of taste, so to speak, although the wiki article indicates it's typically allowed).
https://en.wikipedia.org/wiki/Oscillation_(mathematics)#Oscillation_of_a_sequence
(I know it talks about sequence not series, but the behaviour of a series $(a_n)$ is always just the behaviour of the sequence $(b_n)$ defined by $b_n = \sum_{k=1}^n a_k$.)
Whether you want to trust that the people writing the course are expecting the correct mathematical definition (which requires around a year of university maths to understand) is another matter.

I see, so it’s the same as I observed? But ideally, r=-1 would be the prime example in which it oscillates between two limits.

As you say, it would require a person attending university for a year to understand this, as this is just an access-to-engineering course. I imagine there would be some lay-way in the answer they’re expecting.

To note: the course material has a lot of errors, so I’m not sure how I feel about them knowing what they are even asking, lol