I want to Know what is the importance of knowing if the series diverge or converge in our life " I mean the application of that ."
And how can that be useful in chemistry , because I'm a chemistry student and we take this course _ Calculus II _ as " a specialization requirement " .
We could come up with examples from sex, but you probably want something more traditional, so that you could consider a piece of wood or something like that.
We can talk about weapons to kill vampires that are made from logs or we can talk about sharpening pencils, for instance.
Say you go in a certain angle and you always take the same cut with your instrument.
We want to make sure that the graffiti won't break this time.
For that to happen, a certain amount of graffiti must be left intact.
You can actually digitalize the elements of the problem and solve it using series.
If the series you pick does not converge, you will have a problem because your cuts will not reach the target, clearly the case. If it converges to the desired points, then you get your desired result.
An example (slightly advanced) where the concept is used (Example 10.3) can be found through the attached link.
http://tinyurl.com/gtkz972
thanks all , these answers was really helpful .
but Marcia Ricci Pinheiro I couldn't understand your answer , what's the meaning of " graffiti " ?
Graffiti is Italian art-painting. See wikipedia for it. You can understand Marcia's answer better once you solve few mathematical problems of series convergence and divergence, otherwise it can be tricky to explain.
thanks Vikash . I know this meaning but I thought there is another for it and I hope that I will understand the answer after solve few mathe... etc , like you said .
In fact, your question is far m ore complex than may at first appear. And the answer may not be all that simple.
For starters: often, we have a problem in which a small number appears, call it epsilon. If epsilon is small, then epsilon^2 is yet smaller: that is what it means to be small (with respect to one).
So if we have such a problem, we may want to solve it oteratively. First we determine the effect that is proportional to epsilon, then the one tha is proportional to epilon^2 and so on
For example:
sqrt(1 + epsilon) = 1 + epsilon/2 -epsilon^2/8 + epsilon^3/16
Numerical example:
sqrt(1.001) = 1.000499875062461
That is: 1 + 0.001/2 - 0.000001/8 + 0.000000001/16
This is a very useful general tool.
Now mathematicians, being what they are, like to know whether, if you go on and do this until infinity (and you first need to define what *that* means), whether you still get a meaningful result, and whether it coincides with the exact result. That is what is called convergence.
But convergence is perhaps overrated: if you calculate
1 - 100/1! + 100^2/2! -100^3/3! ....
this converges (to exp(-100)). But, contrarily to the above example, you will never be able to compute anything that way. So something can converge without being useful.
On the other hand, the sum
1 + 2! 0.01 + 3! (0.01)^2 + 4! (0.01)^3...
eventually becomes infinite, but the first few partial sums are a very good estimate of the value of an important function.
So: more important than actual convergence is the fact that you should use these techniques when there is a small number. Then they often work, whether or not you have actual convergence.
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