Question about convergent series/sequences...?

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

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.

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+...

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

Lol! It does!

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

$\displaystyle \sum_{n=1}^{\infty} n$ diverges via the null sequence test since $\displaystyle a_n:=n \rightarrow \infty$ as .

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

$\displaystyle \sum_{n=1}^{\infty} n$ diverges via the null sequence test since $\displaystyle a_n:=n \rightarrow \infty$ 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!

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

What about 1+2+3+...

$\displaystyle \sum_{n=1}^{\infty} n$ diverges via the null sequence test since $\displaystyle a_n:=n \rightarrow \infty$ as .

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