# Question about convergent series/sequences...?

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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

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

Thanks

Posted from TSR Mobile

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#2

jh
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?

(Original post by

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

**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

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?

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#3

**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

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.

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#4

(Original post by

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.

**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.

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#5

(Original post by

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.

**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:

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

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#6

(Original post by

What about 1+2+3+...

**shamika**)What about 1+2+3+...

Spoiler:

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

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#7

(Original post by

Please don't tell me this equals -1/12.

**rayquaza17**)Please don't tell me this equals -1/12.

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#8

(Original post by

Lol! It does!

**shamika**)Lol! It does!

I have seen the result you mentioned before but it relies on some rearrangement which I don't think is valid.

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#9

**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

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A series is the sum of the terms in a sequence.

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#10

(Original post by

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.

**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.

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

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#11

**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.

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#12

(Original post by

I recommend taking analysis and complex analysis courses further, they get really interesting and you do see problems like this!

**TheIrrational**)I recommend taking analysis and complex analysis courses further, they get really interesting and you do see problems like this!

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

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#13

(Original post by

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

In a similar way to 1+2+...=-1/12, 1+1+1+... = -1/2

**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

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#14

**shamika**)

What about 1+2+3+...

Spoiler:

Show

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

**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.

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