Introduction to infinite series

Consider an ordinary deck of 52 cards. Now, try to move the topmost card with your finger such that it doesn't tips away or falls. The maximum point till which you can move that topmost card is half way, You can see the illustration below.

If you carefully analyze the tipping point is at the center of the gravity of the card.

So now, lets go further with top card over hung on the second card lets try to move the second card what is the maximum overhang can you get from both of the cards ?

The answer is to think both the cards has a single unit and try to find the center of gravity of the unit, and its halfway along the unit, which is altogether one and half cards long; so its three- quarters of a card length is overhung from combined effort of two cards.

It is shown in the below figure :

If you now start pushing the third card you can see that you can push only one-sixth of a card length.

The total overhang now is a half (from the top card) plus a quarter (from the second) plus a sixth (from the third). This is a total of eleven twelfths of a card.

Can we get an overhang of more than one card ? The answer is yes, if you push another card the overhung will increase by one-eighth of the card length. So total overhung by four cards will be one and one-twenty-fourth card lengths.

If you keep going the overhung accumulates like this.

for 51 cards pushed you will get a shade of 2.25940659073334.

Why restrict ourselves to 52 cards ? Suppose we had more? A hundred cards? A trillion? lets assume we have unlimited supply of cards.

With 52 cards total overhung was

Since all denominators are even, I can take out one-half as a factor and rewrite this as

for hundred cards the total overhung would be

With a trillion cards the it would be

I can tell you with much confidence that total overhang for hundred cards is around 2.58868875882 and for trillion card sits around 14.10411839041479. A stack of trillion standard card would go most of the way from Earth to the moon.

Just out of curiosity if you have unlimited number of cards, So if you want to get a overhang of hundred cards you would need 405,709,150,012,598 trillion trillion trillion trillion trillion trillion cards.

But we need to understand even though this series grows slowly this harmonic series adds up to infinity.

But we should understand that there was one brave Mathematician Srinivasa Ramanujan who has worked with many divergent series and provided finite answers for the same.

Reference for this article is from the book:

'Prime Obsession' by John Derbyshire