This Sum Amazes Me Every Time

The Alternating Harmonic Series convergence is something you could show in a calc 2 class. Although the alternating harmonic series is conditionally convergent, you may not have known why the sum of alternating harmonic series equals ln2.

Time to look at an amazing alternating harmonic series proof!

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#math #brithemathguy #alternatingharmonicseries

Disclaimer: This video is for entertainment purposes only and should not be considered academic. Though all information is provided in good faith, no warranty of any kind, expressed or implied, is made with regards to the accuracy, validity, reliability, consistency, adequacy, or completeness of this information.

[BriTheMathGuy]

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[hasanjakir]

I understand this prove for the first time. Now, I'll just have to practice a while.

[p_square]

It is also the special case of the Dirichlet Eta Function η(1) and also the x = 1 case of the Mercator Series

[BriTheMathGuy]

Did this video explanation go too fast?

[kinshuksinghania]

That's a very neat way of proving that the sum equals ln 2

[karangupta]

Well i just remembered the Taylor expansion of ln(1 + x) = infinite sum( ((-1)^(n+1))x^n/n) (we use it a lot in calculating limits) which kinda solves the problem in a blink but I really liked that you squeezed in the squeeze theorem ;)

[SlimThrull]

"Oh, good. A short video."
30 seconds later... this is going way too fast!"

[The__Leo]

Okay. That limit of inequality...it blew my mind!

[RafaxDRufus]

I really prefer this more advanced and faster going through content

[dtz_]

for a second I was like wait, this isn't what's promised in the thumbnail, then I remembered what natural log means.

[GameXplayTutorial]

This is so clean! Perfect speed imo :)

[FadkinsDiet]

The integral of x^n over 1+x can easily be computed by reindexing, just let y=1+x, it becomes integral of (y-1)/y with altered bounds

[frankreashore]

Another great video. Huge thanks. When I paused the video I realized that you were not really going too fast. I would have done the same.

[anne]

4:17 did you say "I'll sum you in that one" instead of the usual "I'll see you in that one" 😆.

Love your videos btw!

[jensknudsen]

Inequalities are a pain but sometimes they deliver the goods.

[cpotisch]

Why didn’t you just plug -t into the geometric series, integrate from 0 to x-1, and plug in 2?

[axbs]

A cool little thing I found is that if you write the sum as 1 to inf as ((-1)^(n+1))/(xn), for every positive integer above 1 you put for x, the sum equals to the xth root of 2 (2^(1/x)).

neat.

[xcreeperbombx]

The length of the video is a nice number

[Vapouro]

1:34 "We can pull this sum out of the integral with basically no justification since it's a finite sum."
3:40 takes the limit as n approaches infinity, where n was the bound of the earlier "finite" sum.

... I feel like justification might have been needed.

[alexszczepucha]

Got tired of writing out your math backwards huh 😂