The Taylor Formula презентация

The Taylor Formula
where
For instance,
where

Question 1. If
Solution: Taylor’s formula tells us
for all real numbers x, then

Therefore

Answers to Questions from Light #7:
Taylor’s Formula & Infinite Series
Question 2:
Question 4:
Question 3:

Calculus++
Infinite Series

Question 1. What is the greatest value of b for which any function f that satisfies the properties (i), (ii), and (iii) must also satisfy f (1) < 5?
(i) f (x) is infinitely differentiable for all x;
(ii) f (0) = 1, and
(iii) for all
Solution: Taylor’s formula tells us

Using the properties (ii) and (iii) we obtain
Solve the equation to obtain b = 12.
Therefore, the greatest value of b for which f (1) < 5 is 12.

In particular,
where
That is

Question 2. Use the Taylor formula to show that e is irrational.
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

an integer number
Contradiction!
Thus, our assumption that e is a rational number leads to a contradiction.
Therefore, e is an irrational number.
Therefore

Question 3. Find
Solution. The Taylor formula tells us that for any n:
Therefore
where is a number between 0 and 1.
Multiply by to obtain
= M, an integer number

Since sin x is equivalent to x, when x is small, we obtain
Hence,

Consider an infinite sequence
If we add all the terms of this sequence we obtain an infinite series
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 ?!

The sum of the first n terms of an infinite series, Sn, is called the n-th partial sum of the series
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.

If |b| < 1, the sequence Sn converges.
Example. The geometric 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 diverges (unless a = 0).
If b = –1, the sequence of partial sums Sn also diverges (again, unless a = 0).

A necessary condition for convergence.
If a series
converges, then
Indeed, if the sequence of partial sums
converges, then (Cauchy criterion)
and
Set m = 1, then
Therefore

Question 5. Which of the following series converge?
a. I only b. II only c. III only
d. I and II only e. II and III only
Solution: For series III:
Hence, series III diverges.

a. I only b. II only c. III only
d. I and II only e. II and III only
The sequence cos(k) diverges as
Hence, the sequence
does not converge to 0 as
Therefore, series II diverges.
In series II:

Important series.
This series converges if q > 1, and it diverges if

Question 5. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height rh, where 0 < r < 1.
Suppose that the ball is dropped from an initial height of H meters.
a) Assuming that the ball continues to bounce indefinitely, find the total distance that it travels.
Solution:

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A bouncing ball – total distance travelled

Question 7. A certain ball has the property that each time it falls from a height h onto a hard, level surface, it rebounds to a height r h, where 0 < r < 1.
Suppose that the ball is dropped from an initial height of H meters.
b) Calculate the total time that the ball spends bouncing.
Hint: A ball having zero velocity falls ½ gt2 meters in t seconds.
Solution: Actually, philosophers might find it obvious that the ball never stop bouncing.

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2
A bouncing ball – total bouncing time

Question 7. c) Suppose that each time the ball strikes the surface with velocity v, it rebounds with velocity – kv, where 0 < k < 1.
How long will it take for the ball to come to rest?
Solution:
The velocity of the ball when it hits the ground for the first time is given by
where