Proving that 1/1 + 1/2 + 1/3 + 1/4... approaches INFINITY! #shorts

#shorts #short #harmonicseries #fractions #infinite #infinity #proving #proof #prove #number #numbers #math #mathematics #surprising

Combo Class, taught by Domotro, is a crazy educational show where you can have fun learning rare things about math, science, language, and more! This is the channel for Combo Class SHORTS and EXTRAS - make sure you're also subscribed to the main Combo Class channel where full episodes go!

DISCLAIMER: any use of fire, tools, or other science experiments in this series is always done in a safe and professional way. Do not try to copy any actions you see in this series yourself.

[regulareric]

the weirdness of it stems from how slow it approaches infinity but keeping in mind that you'll never run out of digits makes sense of it.

[dany_]

Therapist: "Homeless mathematician Jack Harlow doesn't exist, he can't hurt you."

_Homeless mathematician Jack Harlow:_

[Domotro]

Note: to people who think "of course any infinite series would approach infinity", that's not true for all infinite series. Imagine the infinite series "1/2 + 1/4 + 1/8 + 1/16 + ..." which just converges toward the number 1 and will never pass it. Whereas "1/1 + 1/2 + 1/3 + 1/4 + ..." will eventually pass every finite size

[wolfgar_m]

A group of Infinite mathematicians walk into a bar. The first one goes: „barkeeper, one beer please“.
The second one goes: „barkeeper, half a beer please.“
The third one goes: „Barkeeper, a quarter beer please.“

The barkeeper puts two beers on the counter and says: „boys, you still don’t know your limit, don’t you?“

…nerdiest joke I know😅

[bungusbongus]

Was confused why it would be considered smaller then I remembered how fractions work lol

[rizsley]

Jack Harlow teaching maths better than our Math teacher lmao 💀

[hunteralexander]

Seeing the harmonic series getting closer together as explained by fractions makes more sense than anything else that I’ve ever experienced in my life

[illunation]

Bro settled himself in a lost place to get that and got crazy too 💀

[shxatterrornotfound]

Little calculus lesson
This series is called the “harmonic series.” Its notable because it is the highest power of p in a p-series that produces a divergent sum.

A p-series looks like this;
sum(1/n^p)
I don’t have a great way to illustrate sun in text, just note that it’s a sum of infinite terms.

If p > 1, then the series converges, meaning it adds up to a finite number.
If p <= 1, then the series diverges, meaning it approaches infinity.

[frivolouss]

Never knew jack harlow was into teaching maths

[edwardwilliamson]

The volume of your voice as you continue speaking approaches infinity

[littlev]

proof by integral test:
integral of 1/x is ln x. lnx approaches infinity. done.

[fyt]

The intermediate values of the series are actually asymptomatically equivalent to a logarithm, which is why it grows so slowly.

[mrcoolst]

Now to confuse some people

Let's say we add this pattern infinitely

1 + (1/2) + (1/4) + (1/8) + (1/16) + (1/32) + (1/64) + (1/128) + •••

This infinite series will actually equal "2"

[eyaph]

I feel like bro knows like... a LOT about dinosaurs

[renatocerrato]

This is the demonstration of Nicole d'Oresme of about 650 years ago much earlier than calculus was actually invented, the name of this sum is harmonic serie and today we demonstrate its divergence in other ways less intuitive

[minz]

dude be looking like an insane time traveler trying to explain to me the concept time and space

[bigfry]

what an amazing way to introduce the comparison test. Beautiful content

[ri-gor]

I really like that proof; very simple and intuitive :)
It reminds me of another proof involving the summation of all fractions where the numerator is 1 and the denominator is a non-negative power of two, but with a totally different solution. I believe that one approaches two interest of infinity. I love how such a small difference on the input can make such a large difference on the output!

[RigoVids]

The basic reason from calculus is that the series is similar to the integral of 1/x, which is equal to the natural logarithm. Since the natural logarithm is the inverse of a function with a complete domain, the log itself has a complete range, meaning it reaches every y value possible. To this end, the natural log must approach infinity, meaning the sum must also approach infinity.
There is also a fun sum where you compare this to powers of (1/2), such that all the pairs of powers add up to 1/2.