The Taylor Formula презентация

Question 4:

Question 3:

Solution: Taylor’s formula tells us

Therefore, the greatest value of b for which f (1) < 5 is 12.

Solution. Let us assume that e is rational,

Then the Taylor formula tells us that for any n:

Take

and multiply both sides

of the double inequality by n! to obtain

Therefore

Therefore

where is a number between 0 and 1.
Multiply by to obtain

= M, an integer number

Hence,

For example, consider the sequence

The corresponding infinite series is

What is the value of this infinite series?
This infinite series does not have a value.

or ?!

An infinite series converges, if converges the sequence of its partial sums:

The limit, S, of the sequence of partial sums is the sum of the infinite series.

The n-th partial sum of the geometric series is given by

If and |b| > 1, the sequence of partial sums Sn diverges.

If a = 0, the sequence Sn converges to 0.

If b = –1, the sequence of partial sums Sn also diverges (again, unless a = 0).

converges, then (Cauchy criterion)

and

Set m = 1, then

Therefore

Solution: For series III:

Hence, series III diverges.

The sequence cos(k) diverges as

Hence, the sequence

does not converge to 0 as

Therefore, series II diverges.

In series II:

Solution:

Solution: Actually, philosophers might find it obvious that the ball never stop bouncing.

Solution:

The velocity of the ball when it hits the ground for the first time is given by

where

In our case

Hence

Therefore

and