Question about convergent series/sequences...?

Watch
A Confused Guy
Badges: 0
Rep:
?
#1
Report Thread starter 6 years ago
#1
Is it always true that a sequence a_{n} converges if and only if \sum_{n=1}^{\infty} a_{n} converges?

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

Thanks


Posted from TSR Mobile
0
reply
davros
  • Study Helper
Badges: 16
Rep:
?
#2
Report 6 years ago
#2
jh
(Original post by A Confused Guy)
Is it always true that a sequence a_{n} converges if and only if \sum_{n=1}^{\infty} a_{n} 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
reply
poorform
Badges: 10
Rep:
?
#3
Report 6 years ago
#3
(Original post by A Confused Guy)
Is it always true that a sequence a_{n} converges if and only if \sum_{n=1}^{\infty} a_{n} converges?

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

Thanks


Posted from TSR Mobile
Certainly not.

Consider the case when \displaystyle a_n=\frac{1}{n}.

We have \displaystyle a_n=\frac{1}{n} \rightarrow 0 as \displaystyle n \rightarrow \infty.

However we also have that \displaystyle \sum_{n=1}^{\infty}\frac{1}{n} diverges. This is the harmonic series.

Note: Sequences are just a list of numbers \displaystyle a_1, a_2, a_3..... strictly speaking it's a function \displaystyle f:\mathbb{N}\rightarrow\mathbb{R}.

Whereas a series is when you add up a terms of a sequence.
0
reply
BlueSam3
Badges: 17
Rep:
?
#4
Report 6 years ago
#4
(Original post by poorform)
Certainly not.

Consider the case when \displaystyle a_n=\frac{1}{n}.

We have \displaystyle a_n=\frac{1}{n} \rightarrow 0 as \displaystyle n \rightarrow \infty.

However we also have that \displaystyle \sum_{n=1}^{\infty}\frac{1}{n} diverges. This is the harmonic series.

Note: Sequences are just a list of numbers \displaystyle a_1, a_2, a_3..... strictly speaking it's a function \displaystyle f:\mathbb{N}\rightarrow\mathbb{R}.

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

Or if you want easier example: take the constant sequence (1). 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
reply
shamika
Badges: 16
Rep:
?
#5
Report 6 years ago
#5
(Original post by BlueSam3)
Or if you want easier example: take the constant sequence (1). 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.
What about 1+2+3+...

Spoiler:
Show
Because you might be surprised that in some sense, that's finite!
0
reply
rayquaza17
Badges: 17
Rep:
?
#6
Report 6 years ago
#6
(Original post by shamika)
What about 1+2+3+...

Spoiler:
Show
Because you might be surprised that in some sense, that's finite!
Please don't tell me this equals -1/12.
0
reply
shamika
Badges: 16
Rep:
?
#7
Report 6 years ago
#7
(Original post by rayquaza17)
Please don't tell me this equals -1/12.
Lol! It does!
0
reply
poorform
Badges: 10
Rep:
?
#8
Report 6 years ago
#8
(Original post by shamika)
Lol! It does!
\displaystyle \sum_{n=1}^{\infty} n diverges via the null sequence test since \displaystyle a_n:=n \rightarrow \infty as \displaystylen n \rightarrow \infty .

I have seen the result you mentioned before but it relies on some rearrangement which I don't think is valid.
0
reply
username1560589
Badges: 20
Rep:
?
#9
Report 6 years ago
#9
(Original post by A Confused Guy)
Is it always true that a sequence a_{n} converges if and only if \sum_{n=1}^{\infty} a_{n} 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
reply
shamika
Badges: 16
Rep:
?
#10
Report 6 years ago
#10
(Original post by poorform)
\displaystyle \sum_{n=1}^{\infty} n diverges via the null sequence test since \displaystyle a_n:=n \rightarrow \infty as \displaystylen n \rightarrow \infty .

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 \zeta (-1).

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
reply
TheIrrational
Badges: 15
Rep:
?
#11
Report 6 years ago
#11
(Original post by poorform)
\displaystyle \sum_{n=1}^{\infty} n diverges via the null sequence test since \displaystyle a_n:=n \rightarrow \infty as \displaystylen n \rightarrow \infty .

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
reply
poorform
Badges: 10
Rep:
?
#12
Report 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
reply
poorform
Badges: 10
Rep:
?
#13
Report 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 \zeta (-1).

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
reply
BlueSam3
Badges: 17
Rep:
?
#14
Report 6 years ago
#14
(Original post by shamika)
What about 1+2+3+...

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

(Original post by poorform)
\displaystyle \sum_{n=1}^{\infty} n diverges via the null sequence test since \displaystyle a_n:=n \rightarrow \infty as \displaystylen n \rightarrow \infty .

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.
0
reply
X

Quick Reply

Attached files
Write a reply...
Reply
new posts
Back
to top
Latest
My Feed

See more of what you like on
The Student Room

You can personalise what you see on TSR. Tell us a little about yourself to get started.

Personalise

Should the school day be extended to help students catch up?

Yes (96)
27.51%
No (253)
72.49%

Watched Threads

View All