PRACTICAL CLASS № 12

Numerical series. The convergence of the sum of the series. Alternating series. Absolute and conditional convergence of the series. Functional series. The area of series convergence. Power series. The radius of convergence

Theoretical questions:

1. Numerical series

2. The convergence of the sum of the series

3. Alternating series

4. Functional series

5. Power series

Classroom assignments:

1. Find at least 10 partial sums of the given series. Is it convergent or divergent? Expalin.

a). ; b). .

2. Determine whether the series is convergent or divergent. Find sum of series if it’s convergent.

a). ; b). .

3. Test the convergence of the series

a) ; b) ; c) ;

4. Find the radius of convergence and interval of convergence of the power series

a) ; b). ; c). .

Homework:

Theoretical material: Basic concepts of probability theory. Addition and multiplication theorems of probabilities.

Solve problems:

1. Determine using integral test whether the series convergent or divergent

a). ; b). ; c). .

2. Test the following series:

a). ; b). .

c) is convergent d) is convergent e) is divergent

f) is divergent g) is divergent h) is convergent (Hint: consider a convergent geometric series with , )

i) is convergent j) is convergent

3. Test the convergence of the series

a) ; b ) ; c ) .

4. Test the convergence of the series

a) ; b) .

5. Apply ratio test to verify that the given series are absolutely convergent and thereby convergent

a) b) c) d)

e)

6. Find a power series representation of the function and determine interval of convergence.

a). ; b). ; c). .

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