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Examples of solving problems for the section series

Lecture



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Example N 1

Investigate a series on convergence Examples of solving problems for the section series .

Decision.

Find

Examples of solving problems for the section series

Since 2> 1, then, on the basis of Cauchy, the series diverges.

Answer: the series diverges.

Example N 2

Investigate on the convergence series Examples of solving problems for the section series .

Decision.

Since factorials are present in U n , we use the dalamber sign.

Examples of solving problems for the section series

Consequently, on the basis of d'Alembert, the series diverges.

Answer: the series diverges.

Example N 3

Investigate on the convergence series Examples of solving problems for the section series .

Decision.

The Cauchy symptom does not work in this case, since Examples of solving problems for the section series . Here we apply the necessary sign of the convergence of series.

Examples of solving problems for the section series

Consequently, on the basis of the required characteristic, the series diverges.

Answer: the series diverges.

Example N 4

Investigate on the convergence series Examples of solving problems for the section series

Decision.

Let's make a series of modules of members of our series:

Examples of solving problems for the section series

we obtain a sign positive series.

Let's apply the limit sign of comparison. Let's make a row equivalent to a row (*):

Examples of solving problems for the section series
Examples of solving problems for the section series

series are equivalent. Because Examples of solving problems for the section series - diverges, the series (*) also diverges. Consequently, the original series does not converge absolutely.

We investigate conditional convergence on the basis of Leibniz:

Examples of solving problems for the section series

therefore, our series converges conditionally.

Answer: the series converges conditionally.

Example N 5

Find the interval of convergence of the series Examples of solving problems for the section series .

Decision.

Completing the replacement Examples of solving problems for the section series get a row Examples of solving problems for the section series . Find the radius of convergence:

Examples of solving problems for the section series ,

Consequently, Examples of solving problems for the section series By performing a reverse replacement, we get Examples of solving problems for the section series . Solve the inequality: Examples of solving problems for the section series , Examples of solving problems for the section series .

Check the convergence of the series on the boundaries of the interval of convergence:

Examples of solving problems for the section series

Consequently, the series converges absolutely when Examples of solving problems for the section series in all other points it diverges.

Answer: the series converges absolutely when Examples of solving problems for the section series in all other points it diverges.

Example N 6

Calculate the integral Examples of solving problems for the section series accurate to 0.001.

Decision.

Using Maclaurin's series of functions Examples of solving problems for the section series :

Examples of solving problems for the section series

Express the function Examples of solving problems for the section series via functional range:

Examples of solving problems for the section series

Since the region of convergence of this series is the interval Examples of solving problems for the section series and spacing Examples of solving problems for the section series , then integrating the obtained series by term, we find:

Examples of solving problems for the section series

Received alternating series, therefore, the error in the calculation of the amount does not exceed the modulus of the first rejected member of the series. Find a member of the series, the value of which is less than 0.001:

Examples of solving problems for the section series

Therefore, to achieve the required accuracy, it is necessary to drop the member of the series Examples of solving problems for the section series and all the following:

Examples of solving problems for the section series
Answer: Examples of solving problems for the section series

created: 2014-09-20
updated: 2023-07-16
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