A little story
One day your boss says, "Great job, we want to give you a bonus, were you planning a holiday this year?". You say "well actually, I wanted to go here", you show him a website, "but I can't afford it". "How much is it?". "1000 euro", you say. "Great, we'll pay for that". "Thanks boss!". Happy days.
Later you realise that if they just give you 1000e you'll have to pay tax on that. The tax rate is 40% so that's a 400e tax bill, you still can't afford the holiday.
You head up to your boss's office and explain the problem. He's still in a great mood and says, "Don't worry, we'll pay your tax bill too". "Wow but there's another bit of a problem, I'll have to pay tax on that extra 400 too. 40% of 400e is 160e, I still can't afford it". You boss is looking less happy now and gets out a piece of paper. "Right, we'll pay all your tax, no matter what" and starts writing down some figures 1000 400 160 64 25.60 10.24 4.096 1.6384 0.65536 you hear him mumbling as he starts adding them all up. Suddenly Joan, his secretary, who's been quiet all this time blurts out "1666e and 66c". You both stare at her in amazement then after a lot more mumbling the boss says "my total is 1666e and .23c but I suppose if I added a few more lines to the sum it'd probably be 1666.66. How did you get it Joan?"
"Well, tax is 40% so he gets to keep 60% of anything you pay him. So the final amount in his pocket is the what you give him times 0.6 . So you're looking for a number that when you multiply it by 0.6 gives you 1000. So 1000 ÷ .6 is the number you're looking for because when you multiply that by 0.6 the two .6s cancel out and only the 1000 is left and 1000 ÷ 0.6 is 1666.666666666...", says Joan.
So what we have proved is that 1000 + 1000 x 0.4 + 1000 x 0.4^2 + 1000 x 0.4^3 + ... = 1000 ÷ 0.6 . There was nothing special about our choice of 40%, so replacing 0.4 with r (and 0.6 by 1-r) and dividing out the 1000 on both sides gives 1 + r + r^2 + r^3 + ... = 1 ÷ (1 - r) Of course there are other ways to prove this but, I like my proof by taxation because it feels like it explains why they are equal.