Calculus++. Find the radius of convergence for 3n презентация

the power series

b) III only

S2 and S3 are divergent

Question 2:

Side length:

Total area:

Total area:

Side length:

Total area:

Side length:

Solution: Let’s try your favourite values

and

For

we have

Converges.

Suppose that

and

where c is a finite positive

If

converges and c = 0, then

If

converges as well.

diverges and

If

then

diverges as well.

Solution: For series III

Consider

Then

Since

diverges, series III diverges as well.

Since the series

converges as well.

For series II

Consider

Then

the series

converges,

Since the geometric series

converges as well.

For series I

Consider

Then

the series

converges

For example, for the series and

the series of absolute values are

series by taking absolute values:

given by and

we say that the series converges absolutely.

series converges, then the series

converges as well.

then we say that the series

converges conditionally.

diverges,

If

where

Solution: For series I:

and the sequence an does not converge to 0.

Thus, the necessary condition of convergence is not satisfied. Hence, series I diverges.

For series III:

Let’s compare |an| with

Hence, series III converges absolutely.

and

The series

converges.

Consider the function

correct

is non-increasing, for n = 1,2,3…

Solutions:

for

Hence, the sequence

decreases for n = 1,2,3…

Thus, the series

converges conditionally.