The Taylor Formula

Сентябрь 13, 2022

Содержание

2. The Taylor Formula where For instance, where
3. Question 1. If Solution: Taylor’s formula tells us for all real numbers x, then
4. Therefore
5. Answers to Questions from Light #7: Taylor’s Formula & Infinite Series Question 2: Question 4: Question
6. Calculus++ Infinite Series
7. Question 1. What is the greatest value of b for which any function f that satisfies
8. Using the properties (ii) and (iii) we obtain Solve the equation to obtain b = 12.
9. In particular, where That is
10. Question 2. Use the Taylor formula to show that e is irrational. Solution. Let us assume
11. an integer number Contradiction! Thus, our assumption that e is a rational number leads to a
12. Question 3. Find Solution. The Taylor formula tells us that for any n: Therefore where is
13. Since sin x is equivalent to x, when x is small, we obtain Hence,
14. Consider an infinite sequence If we add all the terms of this sequence we obtain an
15. The sum of the first n terms of an infinite series, Sn, is called the n-th
16. If |b| Example. The geometric series The n-th partial sum of the geometric series is given
17. If b = 1, the sequence of partial sums Sn diverges (unless a = 0). If
18. A necessary condition for convergence. If a series converges, then Indeed, if the sequence of partial
19. Question 5. Which of the following series converge? a. I only b. II only c. III
20. a. I only b. II only c. III only d. I and II only e. II
21. Important series. This series converges if q > 1, and it diverges if
22. Question 5. A certain ball has the property that each time it falls from a height
23. «Ацкок» 2 A bouncing ball – total distance travelled
24. Question 7. A certain ball has the property that each time it falls from a height
25. «Ацкок» 2 A bouncing ball – total bouncing time
27. Question 7. c) Suppose that each time the ball strikes the surface with velocity v, it
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The Taylor Formula
where
For instance,
where

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

x, then

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Therefore

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

Question 4:

Question 3:

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Calculus++
Infinite Series

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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

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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.

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In particular,
where
That is

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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

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an integer number
Contradiction!
Thus, our assumption that e is a rational number

leads to a contradiction.
Therefore, e is an irrational number.

Therefore

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Question 3. Find
Solution. The Taylor formula tells us that for any

Therefore

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

= M, an integer number

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Since sin x is equivalent to x, when x is small,

we obtain

Hence,

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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 ?!

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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.

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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.

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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).

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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

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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.

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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:

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

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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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«Ацкок»
2
A bouncing ball – total distance travelled

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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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