Convergence of an Interesting Series
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- Опубликовано: 21 июл 2024
- We show that the sum from 1 to infinity of sqrt{n^4 + 1} - n^2 is convergent. We also include a simple proof that the sum of 1/n^2 is convergent.
00:00 A trick to simplify
02:27 Convergence of 1/n^2
04:20 Method of differences
06:29 Considering negatives
There is a slight flaw in argumentation. The fact that the series is bounded from below does not automatically mean that it's convergent. The sine function is bounded too, but it does not converge. The reason why it's convergent is that it is bounded from above and increasing (per your argument that each individual element in the sum is greater than zero).
It is bounded, and every term is positive, so the sum is strictly increasing, and limited from above. That is enough
Very good point! This could have been explained much more clearly - every term is positive, so the important property is that the sum is increasing, rather than just that it is bounded from below.
@@DrBarker we have in 1:40 Sum =1/(sqrt(x^4+1)+x^2)>0
I guess the relevance of positive terms isn't just that the sum can't be negative, but also that the partial sums keep increasing towards either infinity or an upper bound. After all, a general series could also diverge by oscillation.
Exactly
Very good point - this could have been explained much better!
The quick and dirty way would go something like this:
sqrt(n^4 +1) = n^2(sqrt(1 + 1/n^4) ~ n^2(1+1/(2n^4)) = n^2 + 1/(2n^2). The whole expression therefore ~ 1/(2n^2) …
My first idea was multiplying by (sqrt(n^4+1)+n^2)/(sqrt(n^4+1)+n^2)
then comparison test
Series 1/n^2 is convergent so given series also is convergent
Based on that he wrote in the description it is good way
My method too.
Very nice!
Alternatively, set (n+e)^4=n^4+1, from which it follows that 0 < e < 1/4n^3.
Therefore sqrt(n^4+1) -n^2 < (n+ 1/4n^3)^2 - n^2 = 1/2n^2+1/16n^6
Knowing that Σ1/n^k exists for k>1, it follows that the series is convergent.
Using the known values for ζ(2) and ζ(6) it follows that the value is smaller than π^2/12 + π^6/15120 < 0.89
This is a very nice alternative!
Your second and third lines have errors because they are missing required grouping symbols.
For example, 1/4n^3 *means* (1/4)n^3 by the Order of Operations. So, you need to have written
0 < e < 1/(4n^3) to express what you intended. And so on.
Lovely. A nice quick derivation with one eye on the election, I guess.😆
Let (n^2 + eta)^2 = n^4 + 1
It follows that eta^2 + (2n^2)eta = 1
It follows that eta must be smaller than 1/(2 * n^2) if we have (eta^2 + (2n^2)eta) = 1
It can also be noted in passing that eta^2 will be smaller than 1/(4* n^4)
Accordingly sqrt(n^4 + 1) = n^2 + era
So, sqrt(n^4 + 1) - n^2 = eta
And since eta < 1/n^2, the sum of the series must converge.
Of course it's convergent. It's (n²+1/(2n²)+O(1/n⁶)-n²), so its O(n^-2).
3 ton Elephant in the room is the sum of the squares of the reciprocals famously converges to PI^2/6.
Your way is instructive in the sense is proves convergence without finding the value series converges to.
The sum is approximately √2 - 5 +√17- (5/8) + (π^2/12) 😂
Just multiply by root(n^4+1) +n^2 and divide by the same. You get 1 over root(n^4 +1) + n^2 that clearly goes to zero.
That's a necessary but not sufficient condition for the series to converge.
I'm not watching the video. At a glance, multiplying top and bottom by the conjugate yields 1/n^2+more which converges by the p-test.
Somebody hit me up if that's not the end result, or how it was done.
Pls and thnx.
I haven't watched the video but i think comparison with 1/n² would work well
sqrt(3)? Edit: no it's slightly larger than this when N is 200..
Edit again: If you take n from 0 to +inf, the sum is sqrt(3) + some change, approx. sqrt(3.01)
For The Completion!
what about the value?
That is likely very difficult, if not impossible, to figure out. I would bet that it is unknown. Even computing the much more basic sum of the terms 1/n^2 requires somewhat sophisticated machinery.
I still do not get the point of determining convergence without an actual sum. It seems to me like and empty game. Sex without orgasm, food without swalowing, vodka without alcohol...
What is the sum and how to find it? Otherwise it is meaningles in my opinion.
But what is the sum?
That's another beast entirely
I calculated it, its around pi^2 /12
@dalibormaksimovic6399
No. It is around
0.734572122454611
But pi^2/12 is around
0.822467
Percentage error is about 11.97%. Hence not a good approximation.
@@user-cd9dd1mx4n I know, when I just ignored in expansion everything after n^4
@@dalibormaksimovic6399 Why would you ignore, if that will lead to a significant error (above 10%)?
I may ignore insignificant terms, only when my final result has an error 2% at most (depending upon my application).
Generally an error of 10% is huge.
Got lost in the first step😂😂
I checked out by mental calculation and it seems to converge towards zero and it is not too difficult to see why and one could show per complete induction that the larger the number to get the root of the smaller gets your result in that you got always a quadratic number plus one, which shreds you the decimal fractions in ever smaller pieces rooting them, so you end up converging towards zero.
Le p'tit Daniel, Mama Christine I want to be with you making maths and burgers🐕🐕🐕🐕🐕
No, that is not logical.You are already starting out with a positive value in the summation when n = 1, and every term that is added is necessarily positive, so that the sum must be greater than zero.