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.
In: 248
The eli5 is that if an infinite sum adds up to a finite value it’s because at every step in the sequence you add less than the difference between that value and what you’ve added up before. It doesn’t really depend on things being less than a specific number like 1, but it does imply that for any number you pick, an infinite number of terms in the series are going to be less than it. There are many ways to achieve this but the details of exactly determining if the sum is finite (the series converges) is [beyond eli5](https://en.wikipedia.org/wiki/Convergent_series#Convergence_tests). Simply decreasing terms aren’t enough: for example 1/2+1/3+1/4+1/5… doesn’t converge.
Someone showed me this and it made it make sense.
The thought is “if I and an infinite number of things, no matter how small, an infinite number of times it should reach infinity.
So let’s start with the number .9
Let’s add .08=.98
Then .007=.987
.9876
.98765
.987654
.9876543
.98765432
.987654321
.9876543219
.98765432198
Etc
Do this forever! The answer will never reach infinity.
*edit
Dang, I don’t know how to make it to keep my formatting. it looks way cooler when the numbers make a little pyramid
It sounds like you might be thinking of a mathematical concept called a “telescoping series” or a “telescoping sum.” In such a series, most of the terms cancel each other out, leaving only a finite number of terms to compute. This often leads to a simpler expression for the sum of an infinite series.
Here’s an example i found that’s not too advanced: https://courses.lumenlearning.com/calculus2/chapter/telescoping-series/
I would type it out but it would be harder to understand in non-math text
If every term is 1 or more, then yeah, the sum will be infinite. The terms have to get smaller and smaller and/or sometimes be negative for the sum to finite (and even then it’s not guaranteed! 1 + 1/2 + 1/3 + 1/4 + 1/5 + … is infinite!).
For positive stuff it has to get small fast enough. Anything where you get the next term by multiplying by something between 0 and 1 shrinks fast enough (e.g. 1 + 99/100 + (99/100)^2 + … = 100).
It’s entirely possible for *some* of the terms to be > 1 and still have a finite sum. Just multiply everything by a constant, e.g. given that 1+1/2+1/4+… = 2 it should be obvious that 2+1+1/2+… = 4.
Let’s do a really simple one:
1/10+1/100+1/1,000+1/10,000…
In other words: .1 + .01 +.001 +.0001…
What do we get? Obviously 0.1111111… (repeating), which is converges to 1/9.
So in this simple case we can easily see that continually adding an extra 1 after the decimal point has an upper bound. No matter how many times we ad an extra 1 after the decimal, the sum quanity will never even reach 1.2
We can extend this concept intuitively to say generally that if the successive terms shrink sufficiently quickly, they are so many decimal places removed from the previous terms that they can never “catch up”
There are two things happening here: numbers keep getting added, and numbers are being chosen to add. How those numbers are chosen affects how big the total get. If the choice is too big, the total will get bigger forever. However, if the numbers being added are small enough, it will keep changing, but it’ll be pointed towards a particular number.
This is like riding a bike to the store. At first, if you just go in the general direction of the store, you are getting closer, but as the distance gets shorter, you need to make smaller, more accurate adjustments to keep yourself on track.
This can go on “forever”, making smaller and smaller adjustments. At least until it’s too hard to stay upright on your bike. Then you get off it, and go inside.
People call heading towards the goal place or number **converging**, (*Latin: turning towards*). They call heading away **diverging** (*Latin: turning away*).
There’s a subtlety that is being missed in most answers here. An “infinite sum” is not the same thing as a finite sum. I.e. 1+1/2+1/4+… is not the same kind of thing as 1+2+3. It doesn’t follow from the normal rules of addition that one learns in primary school that the infinite sum just given sums to 2.
Instead, what happens is that mathematicians *define* an “infinite sum”, using some moderately sophisticated maths to specify what the actual value of the sum will be when you evaluate it. Of course, they choose sensible rules so that the outcome makes some intuitive sense. See other answers for the actual intuition in play.
Think of a situation where you are in a room. You can take one step at a time, but each step is half of the distance between where you are and the other side of the room. It is important to note: another way to say this is your first step is half of the room, and every step after is half the distance of the previous step but we will come back to that. Now, mathematically speaking, since you are always getting halfway between where you are and the end, you will never reach the end, but you will keep getting closer and closer to a single spot no matter how many times you step. So an infinite number of distances still converge to a single distance value.
Now this only works if the absolute value of the change in terms generally gets smaller over time. (this only works if you are moving towards a spot, sometimes you can go past the spot, if you are guaranteed to go back and forth over the spot getting closer and closer each time) for example, imagine you are on a football field standing at one end. Every step you take is 1.5 times the distance between you and the 50 yard line in the direction of the 50 yard line. You would go from 0, to 75, to 37.5, etc. You would keep going back and forth over the 50 but each step you would be stopping closer and closer to the 50.
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.
This is why a 10 yo invented Calculus. He was tired of getting in trouble for hitting his 8 yo brother who kept saying “I’m not touching you” while continuing to move his finger 1/2 the distance closer to his face. Technically his little brother was right but everyone has limits.
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