series integration

[Ecclesiastes]

I am teaching myself.

I have the Stewart book on Calculus 4th Ed. and I'm having a motivation problem.

I understand that the ability to examine sequences and series would be a helpful thing, but all the series I am seeing are generated by formula. I have the answer already, why am I posing the question with a series?

I also understand that contrived exercises prepare me for real problems. I'm just having a problem seeing the path from where I am to where I am going.

Can you describe for me the way and the reasons for the path from where I am to, say, the Fourier Transformation? I saw that it was a way to resolve a series into a set of waveforms and that there was a new way to perform Fast Fourier Transforms that would allow more efficent compression and transmission of images.

 

[tkhunny]

Series are another tool to express relationships. Using series with closed expression alternatives gives you the opportunity to become acquainted with them.

Series are another tool to turn impractical formulae into useful approximations.

Make sure you know the difference between Divergent, Convergent, Conditionally Convergent, and Absolutely Convergent.

 

[Deleted member 4993]

Values of functions like sin(x), cos(x), tan(x), e^x, etc. are calculated by calculator (or computers) through series approximations.

 

[Ecclesiastes]

math is hard

Values of functions like sin(x), cos(x), tan(x), e^x, etc. are calculated by calculator (or computers) through series approximations.

Series are another tool to turn impractical formulae into useful approximations.

That's a step in the right direction.

I can see that calculation of a series will yield values I want.

I'm reading the Wikipedia article on the use of Madhava's method, but it says how he discovered the series is unknown. I'm sure working through his method will be enlightening, showing me how he considered the problem.

I feel so stupid just now. Bear with me.

We have a series of equations, which define a function.

So, to construct such a series - from scratch, not from the function - I consider a model of the problem roughly and then less so, perhaps by half, stepping towards the limit. This will give me a sequence of equations as terms. I look to find a pattern of change in the terms and take it to define an infinitely extensible sequence I can sum, thus the series. That is if I can find such a pattern that tracks the sequence observed, not modeled.

These ancient mathematicians measured arc-lengths. Elder physicists took measures of pressure along a constriction in a tube. Modern digital algorithms take samples of frequency intensities of a sound.

Sun's up. Back to painting. Later.

 

[Ecclesiastes]

Not quite

I was wrong. I have a series which approximates a continuous function.

Could I get someone to critique what I have so far?

 

[Deleted member 4993]

I was wrong. I have a series which approximates a continuous function.

Could I get someone to critique what I have so far?

I do not understand your statement.....

When you are "approximating" a function with a series, you are most probably looking at an infinite series. Like:

sin(x) = x - x³/3! + x⁵/5! - x⁷/7! ....

so sin(x) is often approximated by:

sin(x) = x - x³/3! (within a restricted domain - when x is relatively small. When x < ± 4°, we can use only 'sin(x) = x ' incurring small error)

This "finite" series ( = x - x³/3!) is a continuous function and approximates a continuous function [sin(x)]

Now what is your question?