SERIES
SERIES CONVERGENCE
THE DIFFERENCE BETWEEN NECESSARY AND SUFFICIENT CONDITIONS.
The first step to finding out if an infinite series converges is to find out if the sequence converges to zero. In other words, the function must approach zero. This is a necessary condition if the sequences approach any number other than zero the series cannot converge. This can be tested with the nth term test. The sufficient condition is the speed in which the sequence approaches zero. If the sequence approaches zero fast enough, the infinite series will converge to a finite value.
WHAT DOES CONVERGENCE/DIVERGENCE MEAN?
An infinite alternating series of numbers to diverge.
An infinite and alternating series of numbers diverges when the sum does not settle on one finite value. This can be tested by seeing if the corresponding positive series converges or diverges.
An infinite positive series of numbers to converge.
An infinite positive series of numbers converges when the sum adds up to a finite number. In order for a series to converge the sequence must approach zero, and must approach zero fast enough so that the sum does not add to infinity.
Power and Taylor Series
Why Taylor Series are so important
Taylor series are incredible. They allow us to approximate functions that we cannot compute. In the most simple definition, a Taylor Series is a representation of a function as an infinite sum. They can be used in many different aspects of our world to approximate very difficult problems we face.
Examples of Taylor Series
#1
#2
Example with VA
Taylor Polynomial
What a Taylor polynomial can do
Taylor Polynomials are even more useful than Taylor series. Taylor Polynomials allow us to approximate curves for whatever functions we can derive. This is especially useful in the engineering world because it allows us to change any function into a polynomial that is much easier to work with. Computers use Taylor polynomials to approximate almost all of the more complex numbers(pi, e...).To compute a Taylor polynomial, Follow the pattern of Taylor series and stop at a finite number of values. Error bounds are very useful and fairly straight forward to compute. Error bounds help us to know how accurate our approximation is, and what value we should stop at to get the desired error.
sin(x) each n term approximates the curve better and better.
Graphs
f(x) = 1/(1-x)
f(x) = e^x
Real-World Example
COMPUTER SCIENCE
Computers need polynomials to represent complex numbers. You are building a new programming language and you need to create an algorithm that can accurately create the number e within 0.001 accuracy. How would you do this?
This question changed the way I think about Taylor polynomials. We use technology every day, and Taylor series play a major roll in the effortless transition between numbers and functions in general. This question was meaningful for me on many levels. I am studying computer science and this understanding of Taylor polynomials will help me create software.
