In my lecture notes, a telescoping sum is a sum where the terms $u_k$ can be written as $u_k=t_k-t_{k-1}$ for a sequence $\{t_k\}$. Consider the terms $u_k=\frac{1}{j^2-1}=\frac12\left(\frac{1}{j-1}-\frac{1}{j+1}\right)$. This is a telescoping sum, however, I can't seem to find a way to write it on the form $t_k-t_{k-1}$ for some sequence $\{t_k\}$.

oh, that's not a very good definition. for one thing, for any sequence a_n whatsoever, a_0 + a_1 + a_2 + ... + a_N = s_0 + (s_1 - s_0) + (s_2 - s_1) + ... + (s_N - s_{N-1}), where s_n = sum_{k=0}^n a_k.. whatever "telescoping" is, is more about having nice formulas or expressions for a sequence of partial sums, and 'cancellation' that occurs in those.

and not so much about the sequence of numbers.

Your letters are totally messed up in that problem, too.

it also kinda misleads as to the concept. sometimes you can have this more general 'telescoping' in ways that take advantage of cancellation with something other than what's in the immediately preceding term.

I agree with Leslie. To fit your definition, you need to write the series as the sum of an even series and an odd series.

ignoring the 1/2 for the moment, write out stuff like (1/1 - 1/3) + (1/2 - 1/4) + (1/3 - 1/5) + (1/4 - 1/6), just informally speaking it seems clear that the cancellation of interest is going to be happening with stuff "two terms back," not one.

so even a nice 'telescoped expression' for the nth partial sum will have one or two more terms than you might expect, reflecting that.

Yes, sorry, replace $j$ with $k$ in my post.

the nth partial sum will look like [some stuff from the beginning] minus [looks like two terms from the end that haven't yet canceled with anything 'yet']

if that makes any sense. and you can imagine similar higher-order examples of cancellation, where even the 'nice' canceled formula for the nth partial sum, though it might have a limit that is easy to evaluate, and though it does not have dot dot dots in it, will still look like shit as a function of n.

Ok, so hitherto my strategy has always been to find a sequence to rewrite the terms of the telescoping sum into (which has worked in many cases), but I guess then I should start simply by writing out the first and last n terms of the telescoping sum

yeah, it ultimately amounts the same thing. (your sequence "t_k" up there is exactly u_1 + u_2 + ... + u_k, at least up to a number that doesn't depend on k)

if you're super into sigma notation you could probably arrive at a nicer formula without ever writing "..." by breaking it up a bit and doing some index shifting, but its maybe conceptually clearer if you just work with exemplary terms to arrive at the formula, and then (if you care) prove it by induction or by algebraic rearrangement (which is easier once you know what the formula is)

let $\phi$ denote the density of a $N(0,I_d)$ random variable.

what is $I_d$

variance?

@geocalc33 No idea. I would suggest reading whatever text you see this notation in, and seeing if it is defined.

I would guess either variance or standard deviation.

yeah, i'd guess variance too, but it isn't exactly something we can vote on or 'figure out'

or if it is i vote that it's the uniform distribution on {0,1}

For reference, in my Google bubble, the first two results for "notation for a normally distributed variable" are variance and standard deviation.

i vote for a coin toss

I forgot to include that $\Bbb R^d$

$N(0,I_d)$

I'll just keep in the back of my mind that it's either variance or standard deviation

or a coin toss

I'll go with variance

@geocalc33 ...

it's variance now

Don't guess. Read your text.

I am quite certain that the notation is defined somewhere.

no, let us vote on it

it's easier to suggest solutions than it is to propose problems, xander

or the other way around. i forget which way the good way is

@leslietownes Wikipedia defines a telescoping sum as a sum whose terms can be written as the difference of a sequence. It's strange then that $u_j=\left(\frac{1}{j-1}-\frac{1}{j+1}\right)$ is an exception to that

schn, this would be an example of wikipedia being trash

:)

i mean, literally every sum is 'telescoping' by that definition. it's not capturing the relevant thing at all

i'm not saying that i'd know what to put in that entry, but that ain't it

@schn That isn't quite the definition Wikipedia gives...

Wikipedia's definition is that a telescoping series is one of the form $\sum t_n$, where there is some sequence $a = (a_n)$ such that $t_n = a_n - a_{n+1}$.

Which is not to say that Wikipedia isn't trash; just that it doesn't seem to far off in this case.

@XanderHenderson is $a_{n+1}$ correct or is it a typo in the article?

@schn Seems fine to me.

The idea is that parts of consecutive terms cancel out.

$$\sum_{k=1}^{n} t_n = (a_1 - a_2) + (a_2 - a_3) + (a_3 - a_4) + \dotsb + (a_{n} - a_{n+1}) = a_1 - a_{n+1}. $$

so, a telescoping series is any series. got it.

that's what i meant. we all know what it's trying to say.

@leslietownes I'm not sure that I agree. It is a question of the way in which the series is presented. A telescoping series is one which is written as $\sum (a_n - a_{n+1})$, not a series which can be written in this form.

@XanderHenderson but you obtained $a_1$ instead of $a_n$, didn't you?

the vibe is right, but it should have been presented as a vibe, not as a definition.

@leslietownes Perhaps.

xander: the page literally says "can be written," which is what you are objecting to. which is also what i'm objecting to.

@leslietownes Ah, yes.

It is trash.

im gonna find the user page of whoever put that there and light it up.

but first, i need a wikipedia account, maybe in a fictitious name, let's say, shed tifrin.

@leslietownes I fixed it.

We'll see how long until it is reverted.

keep me posted. i can unleash a hell of wikipedia bots to edit war over this if need be. the living will envy the dead.

@leslietownes Yay!

I still miss the reason it was wrong ….

"can be" vs "is".

i'd phrase it as, the whole 'telescoping' thing is maybe a property of a symbolic representation of a sequence. it's not a property of the sequence itself. and to be less formalistic - zooming too closely in on "each term cancels with the exact last" is also the wrong intuition about where you are likely to find telescoping.

@leslietownes Yeah, that was my point above.

yeah. and its more of a vibe than a definition. that's OK! no need for some weird cite to a paper from 1644.

which is actually something i like about the page in its current form.

also the bitmap graphic with a bunch of colored text in it. thats like 'classic' math wikipedia to me.

all we're missing is one of those rainbow plots of a complex valued function on the complex plane.

@XanderHenderson OK, I’m being dumb. What series that is not in fact telescoping can be written in that form?

Oh, I see.

i may have goofed on indices or something there, but any sequence is telescoping if you have recourse to its sequence of partial sums.

its more, having a useful expression for that sequence.

Covid isn’t enhancing my brain.

you feeling ok? how long has it been? when i got it, i tested positive for almost 2 weeks.

Interesting, because the way I would define telescoping is in terms of partial sums, anyway, I suppose.

in a way it's a version of that thing in some calculus books, is a "series" something other than a sequence of partial sums. and every sequence is a sequence of partial sums. is a series its value or not. etc.

My roommate got it more than a week ago. I started with sinus stuff on Thursday, tested positive Sunday.

After consulting with the PA today, I decided not to do Paxlovid — probably too late anyway.

hey @Ted @lesie

Hi, Lukas.

howdy

I never thought about a formal definition of telescoping sum before

it seems intuitively very clear what it means

Yeah, it’s challenging

but it's difficult to decide when that applies

probably this definition of telescoping sum is too broad: the sequence of partial sums is periodic

No, that’s not right :)

you can maybe give restrictive definitions in a limited world. for the example that started this all, the sequence of partial sums happens to be a rational function of n, when in general it might not be.

and wilf and zeilberger and whoever have all of those algorithms for when recursion like things happen to involve polynomials or rational functions. maybe in that realm you can formalize something that comes close.