Originally published at: https://boingboing.net/2019/04/02/three-interesting-infinite-ser.html

…

# Three interesting infinite series

No!

Did he say the sum of the first 50T terms is more than 50T? It’s not! If memory serves me, it’s about ln(50T), which would be about… 30 I think?

The second is 1 + 1/2 + 1/3 + 1/4 + 1/5 … which equals infinity

Based on nothing but a very limited knowledge of mathematics and an instant visceral reaction, I’d say that can’t possibly be right!!

ETA: damnit I keep forgetting that the sarcasm tag is ALWAYS necessary!

The series 1 + 1/2 + 1/4 + 1/8 + 1/16 … can be shown to converge on 2 as the number of terms approaches infinity, and will never exceed 2 under any circumstance.

The series 1 + 1/2 + 1/3 + 1/4 + 1/5 … can be shown to **not** converge on any number. Put another way, no matter how large a target number you set, the sum of terms of the series will eventually exceed it, if you just keep adding terms.

Yea, they might want to ask Zeno about that one…

It DOES diverge to infinity, albeit quite slowly. I haven’t looked at the video yet, but a good way to see this is to group the series by powers of 2. For example, the group 1/2 + 1/3 > 1/4 + 1/4 = 1/2, and the next group 1/4 + 1/5 + 1/6 + 1/7 > 1/8 + 1/8 + 1/8 + 1/8 = 1/2 also. This happens for every group that starting with a power of 2 denominator up to the next one, so the series is > 1 + 1/2 + 1/2 + 1/2 + … .

There is a fun short story by PKD which incorporates this paradox, called The Indefatigable Frog. I’ve actually been trying to write my own story, expanding upon his premise to quite a silly degree

Gives me shivers thinking back to writing 8080 assembly code to calculate trig functions using the Taylor series. My math skill probably wasn’t up to the task but sometimes, if you don’t know what you don’t know and are stubborn enough, you can work it all out.

Then you’ll enjoy the video!

Mentioned in the video.

Spelled out in the video.

Not presently in a location conducive to video viewing. Reply based on text in original post.

It’s almost like nobody watched the video before commenting…

(Full disclosure: I haven’t watched the video.)

I stopped watching when he completely failed to explain why the first series converges on 2.

I’m usually a tv;dw;. I often go to the comments, though, in case someone there changes my mind.

For any value less than two, the sum of a finite number of terms in the series will be greater than that value. For any value greater than two, no matter how many terms you add, the sum will never reach that value.

That wasn’t explicitly stated in the video, but the approach used to explain why the second series doesn’t converge hints at it well enough.

The ending was a bit of a cop-out though! He explained quite well that the third series is bounded between 1 and 2, but how the heck did π get involved!?

That’s the basic idea; the natural logarithm and the harmonic series asymptotically differ by the Euler-Mascheroni constant. I think 30 is a little low though. Looks to be about 32.

Pretty good, considering the oriface I pulled it from.

Zeno didn’t know about the quantification of reality - Achilles just teleports across the last plank length (1.6 x10-35 m) to catch the tortoise.

However, I think that he was trying to say that there is some N for which H_N = \sum_{k=1}^{N} 1/k would eventually exceed 50 trillion, not that H_{50 trillion} > 50 trillion. That’s what it’s supposed to be anyway.

I laughed that the third series was a “famous unsolved problem of the 18th century.”

Didn’t any of those dotards take pre-calc?

He used a comparison to the first series to show it converged, he didn’t actually show what it converged to. In the 18th century they did know the series converged using the same argument given in the video, but not what it converged to. I don’t know what the original proof was, but the only proof I know of uses Fourier series, which is somewhat beyond first year calculus, much less precalculus.