Here, if it does converge is going to be greater than zero. Positive we know that this sum right over To one ninth to one sixteenth, and so they're quickly approaching zero, which makes us feel pretty good that this thing has a chance of converging. It looks like they'reĭecreasing quite quickly here from one to one fourth Ninth, plus one sixteenth and it goes on and on and on forever.
Squared, plus one over three squared, which is one To one plus one fourth, that's one over two For example, 1/n diverges, 1/n² does not.Ī bit the infinite series from n equals one to infinity * This behavior only happens if each successive term is small enough. You can get as close as you want to d, but each time you are almost there, you can only travel half the remaining distance, thus the series limit, d, is forever beyond your reach. So even though you are continuously adding to the sum, the values are getting so small that you will never fully cover the distance, d. Each successive term of the series goes a "sufficient distance less"* than that of its predecessor that it is just like the example of moving half the distance more each time. Well to do so you would first need to travel half the distance to d first, but to travel half the distance to d you would first have to travel "half of the half" of the distance to d, and then "half of the half of the half" This argument goes on forever, so it appears that the distance d cannot be traveled. Suppose you wish to travel a distance, d. Zeno argued that motion was inherently impossible. Here is the general idea, and while the example I am about to use is not the sequence above, the same logic applies. But still you wonder, how can adding positive numbers to a sum ever stop the sum from growing bigger and bigger and bigger. So you can see that, yes, the terms are positive, and the are getting smaller quite fast. So what can we say about that? Let’s take a look at the behavior of 1/n². The sum isn't decreasing, but each successive term in the series is.
I hope this is an answer to your question (as I understood it to be):
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