does this apply only for terms that are less than 1 eg 1 + 1/2 + 1/4…. or does this apply to all ap/gp. I remember studying this, but it’s been so long I remember only the gist.
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Place two cakes to your left. Take 1 of them and move it to your right. Take half of the cake from your left to the right (so now there’s 1.5 cakes to your right), then take half of what’s remaining from the left to the right repeatedly.
You’ll notice two things:
– the amount of cake to your right will never exceed two.
– the amount of cake to your right will get closer and closer to two. Mathematicians would say that whatever number just below 2 you pick (such as 1.9, 1.99 or 1.99999), if you repeatedly move half of the remaining cake from the left to the right, you’ll eventually have more cake on the right side than the number you picked.
In such a case, mathematicians just say that this “infinite sum” 1 + 1/2 + 1/4 … equals 2. Strictly theoretically speaking, this isn’t entirely accurate since “infinite sums” don’t really exist.
1 + 1/2 + 1/3 + 1/4 … is a prime example of an infinite sum that doesn’t equal any number (and instead, only grows larger and larger towards Infinity). The simplest proof for this that I know is that obviously, 1/2 is >= 1/2, both 1/3 and 1/4 are >= 1/4, 1/5 to 1/8 are >= 1/8, 1/9 to 1/16 are >= 1/16 and so on. So, you’ve got one number in the sum that is >= 1/2, twice as many numbers that are >= 1/4 (which is half of 1/2), twice as many that are >= 1/8 (which is again half of 1/4) and so on. This means that there are infinitely many groups of numbers in the sum whose sum is >= 1/2 (for example, since 1/5 to 1/8 are all >= 1/8, it follows that 1/5+1/6+1/7+1/8 >= 1/8+1/8+1/8+1/8 = 1/2). The sum of infinitely many 1/2 equals infinity.
Infinite sums get a lot weirder if you put in negative numbers as well. They then lose commutativity for example, which means that if you mix up some of the numbers in the infinite sum, the result will be different.
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