Series - Alternating Series - Power Series - Differentiation and Integration of Power Series

Selasa, 11 Agustus 2020

•        The Alternating Serie Test (Leibniz’s Theorem)

Let {a_n} be a sequence of positive numbers such that 

a_{n + 1} < a_n for all n.

Then the alternating seriesandboth converge.

 

•        Absolute Convergence

•  A seriesis absolutely convergent if the series 

   is convergent.

•  If the seriesis absolutely convergent then it is convergent.

 

•        Conditional Convergence

A seriesis conditionally convergent if the series is convergent but is not absolutely convergent.

 

 

Power Series

 

Real numbers: x, x₀

Power series:

Whole number: n

Radius of Convergence: R

 

•        Power Series in x

   

•        Power Series in (x – x₀)

   

 

•        Interval of Convergence

The set of those values of x for which the function

is convergent is called the interval of convergence.

 

•        Radius of Convergence

If the interval of convergence is (x₀ – R, x₀ + R) for some R ≥ 0, the R is called the radius of convergence. It is given as

   .

 

Differentiation and Integration of Power Series

 

Continuous function: f(x)

Power series: 

Whole number: n

Radius of Convergence: R

 

•        Differentiation of Power Series

Let
 for |x| < R.

Then, for |x| < R, f(x) is continuous, the derivative f’(x) exists and

   

 

•        Integration of Power Series

Letfor |x| < R.

Then, for |x| < R, f(x) is indefinite integralexists and

   

 

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Labels: Mathematician

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