Introduction.
What do we mean When we say series Taylor series, or Taylor expansion around zero or around one or around two. What do we mean? When we say McLaren series,
Part one.
We Define a series as the the sum of X to the power K where K goes from 0 to Infinity. This is a Taylor series around zero. This series is convergent according to the ratio test. If x, the absolute value of x is less than |x| less than 1.
We Define the Radius of convergence The limit as n goes to Infinity of an+1 Over an in absolute value. This is the ratio test for series. This will give us the radius of convergence.
And from there, we can get the interval of convergence.
Part three
When we are inside the radius of convergence, we can differentiate the series. We can also integrate this series in this case, the integrated series or the differentiated series have the same radius of convergence.
We would have to prove these result Because we're going to need them when we are dealing with differential equations.
Part four:
We can use the fact that the differentiated series and the integrated series have the same radius of conversions or the same interval of convergence To find It Expansion series by integrating or differentiating. This is one of the keys that we would have to use when dealing with differential equations title series or MacLaurin series. In this case series are very important in dealing with the differential equations.
Part five:
Applications and examples. In this part, we're gonna have some examples. And also we're gonna work some problems. And we're going to see how to expand the series 1/(1+x^2) squared. When we take it from 1/(1-x) . So we now the expansion of 1 over 1 minus X. How can we get one over Plus x squared? Also, we can integrate to get the arctangent of X from 1 over 1 plus x squared. This case, this example shows, that we can get the series When we integrate or differentiate.
Part 6:
Why do we need to series? As we will see, in the next few videos, we need them to solve differential equations And to find limits. Sometimes we can use l'hospital's rule And therefore, we can use the series expansion to find the series that we need. This is very good way. You need to have a complete Mastery
Part 7
What do I need to make series or to compute series? You need to have a strong algebra. You will also need To know how to integrate, how to differentiate. And also, you need to now series And the definition of improper integral to be able to find unsolved differential equations, based on expansions of series.
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