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#1
Is it always true that a sequence converges if and only if converges?

Also, what is the difference between a sequence and a series?

Thanks

Posted from TSR Mobile
0
6 years ago
#2
jh
(Original post by A Confused Guy)
Is it always true that a sequence converges if and only if converges?

Also, what is the difference between a sequence and a series?

Thanks

Posted from TSR Mobile
The sequence is basically just your list of a_n terms. The series is what you get when you put a summation sign in front of it. Do you not have lecture notes or a standard textbook that gives you the definitions of these?

For your first question, think about a_n = 1/n. Does a_n converge as a sequence as n->infinity? Does the corresponding series converge?
0
6 years ago
#3
(Original post by A Confused Guy)
Is it always true that a sequence converges if and only if converges?

Also, what is the difference between a sequence and a series?

Thanks

Posted from TSR Mobile
Certainly not.

Consider the case when .

We have as .

However we also have that diverges. This is the harmonic series.

Note: Sequences are just a list of numbers strictly speaking it's a function .

Whereas a series is when you add up a terms of a sequence.
0
6 years ago
#4
(Original post by poorform)
Certainly not.

Consider the case when .

We have as .

However we also have that diverges. This is the harmonic series.

Note: Sequences are just a list of numbers strictly speaking it's a function .

Whereas a series is when you add up a terms of a sequence.

Or if you want easier example: take the constant sequence . It's very clear that this converges to 1, but you're going to have a hard time persuading me that adding up infinitely many 1s will come to a finite number.
1
6 years ago
#5
(Original post by BlueSam3)
Or if you want easier example: take the constant sequence . It's very clear that this converges to 1, but you're going to have a hard time persuading me that adding up infinitely many 1s will come to a finite number.

Spoiler:
Show
Because you might be surprised that in some sense, that's finite!
0
6 years ago
#6
(Original post by shamika)

Spoiler:
Show
Because you might be surprised that in some sense, that's finite!
Please don't tell me this equals -1/12.
0
6 years ago
#7
(Original post by rayquaza17)
Please don't tell me this equals -1/12.
Lol! It does!
0
6 years ago
#8
(Original post by shamika)
Lol! It does!
diverges via the null sequence test since as .

I have seen the result you mentioned before but it relies on some rearrangement which I don't think is valid.
0
6 years ago
#9
(Original post by A Confused Guy)
Is it always true that a sequence converges if and only if converges?

Also, what is the difference between a sequence and a series?

Thanks

Posted from TSR Mobile
If.

A series is the sum of the terms in a sequence.
0
6 years ago
#10
(Original post by poorform)
diverges via the null sequence test since as .

I have seen the result you mentioned before but it relies on some rearrangement which I don't think is valid.
Warning: if you've only done a first course in analysis, I wouldn't bother reading the rest of this post.

The easiest (rigorous) derivation is via the analytic continuation of the Riemann Zeta function and then evaluating .

For a much longer discussion, see here. The other main source of derivations comes from advanced physics, where setting the sum as -1/12 makes things like string theory self-consistent. Note that what these methods are not saying is that the analysis you've learnt is wrong and that these divergent series are suddenly convergent. What these methods actually show you is that you can set the value of these divergent series to a finite number and that somehow this makes sense in some rigorous mathematical context. There's actually a lot of theory around divergent series where alternative definitions of summation assign lots of divergent series (in the traditional epsilon-delta definition) with finite values.

In a similar way to 1+2+...=-1/12, 1+1+1+... = -1/2
0
6 years ago
#11
(Original post by poorform)
diverges via the null sequence test since as .

I have seen the result you mentioned before but it relies on some rearrangement which I don't think is valid.
I recommend taking analysis and complex analysis courses further, they get really interesting and you do see problems like this!
0
6 years ago
#12
(Original post by TheIrrational)
I recommend taking analysis and complex analysis courses further, they get really interesting and you do see problems like this!
Cool!

And yeah I will be doing more (complex) analysis courses next year and probably the year after. It's probably my favourite most interesting topic out of all the modules I have done so far (1st year)

thanks
0
6 years ago
#13
(Original post by shamika)
Warning: if you've only done a first course in analysis, I wouldn't bother reading the rest of this post.

The easiest (rigorous) derivation is via the analytic continuation of the Riemann Zeta function and then evaluating .

For a much longer discussion, see here. The other main source of derivations comes from advanced physics, where setting the sum as -1/12 makes things like string theory self-consistent. Note that what these methods are not saying is that the analysis you've learnt is wrong and that these divergent series are suddenly convergent. What these methods actually show you is that you can set the value of these divergent series to a finite number and that somehow this makes sense in some rigorous mathematical context. There's actually a lot of theory around divergent series where alternative definitions of summation assign lots of divergent series (in the traditional epsilon-delta definition) with finite values.

In a similar way to 1+2+...=-1/12, 1+1+1+... = -1/2
Hmmmm seems interesting but way beyond me at the moment. thanks for the insight.
0
6 years ago
#14
(Original post by shamika)

Spoiler:
Show
Because you might be surprised that in some sense, that's finite!

(Original post by poorform)
diverges via the null sequence test since as .

I have seen the result you mentioned before but it relies on some rearrangement which I don't think is valid.
There's two different (but related) ways that I know of to evaluate silly sums like that: one is the complex analysis method noted above, and redefining what we mean by "summation" through abelian means.
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