2.7.3 Practice set: Taylor series

Taylor polynomials

Taylor polynomials (lesson 2.7.1)

0 of 7 correct

1.
Write the degree-2 Taylor polynomial of $e^{2x}$ about 0
2.
The value, slope and second derivative of $\sqrt{x}$ at 4 are 2, $\frac{1}{4}$ and $-\frac{1}{32}$ . Use the degree-2 copy about 4 to estimate $\sqrt{4.2}$
3.
What is the coefficient of $(x - 1)^3$ in the Taylor polynomial of $x^4$ about 1?
4.
Use the degree-2 copy of $\cos x$ about $\pi$ to estimate $\cos(\pi + 0.1)$
5.
Near $a$ , a degree-3 copy misses by 0.0008 at a distance 0.4 from $a$ . Estimate its error at a distance 0.2
6.
Write the degree-2 Taylor polynomial of $\ln x$ about 2. Type ln(2) for $\ln 2$
7.
Write the degree-2 Taylor polynomial of $\sin x$ about $\pi$ . Type pi for $\pi$

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Standard series and the error

Standard series and the size of the error (lesson 2.7.2)

0 of 8 correct

1.
Use the first two terms of the series for $\cos x$ to estimate $\cos 0.2$
2.
Put $-x$ into the series for $e^x$ . Write the result up to the $x^3$ term
3.
Put $-x$ into the series for $\frac{1}{1 - x}$ . Write the result up to the $x^3$ term
4.
$\sin 0.1 \approx 0.1$ . Estimate the size of its error with the first left-out term
5.
$\cos 0.5 \approx 1 - \frac{0.5^2}{2}$ is also the degree-3 copy, since the $x^3$ term is 0. Bound its error with $M = 1$
6.
After degree $n$ , the series for $e^{0.5}$ is within $\frac{2 \times 0.5^{n+1}}{(n + 1)!}$ of $e^{0.5}$ . What is the smallest $n$ that makes this less than $10^{-6}$ ?
7.
The series for $\ln(1 + x)$ converges at one end of $-1 \le x \le 1$ and diverges at the other. At which end does it diverge?
8.
At $x = -1$ the partial sums of $1 + x + x^2 + \cdots$ go 1, 0, 1, 0, … What value does the formula $\frac{1}{1 - x}$ give there?

How to type answers

Powers with ^: x^7, 2^(1/3), a^-2.
Multiply with * or by writing side by side: 3x^2, 2(x+1).
ln(x) is the natural log, log(x, 2) is log base 2, e is Euler's number, sqrt(x) is a square root.
sin(x), cos(x) and tan(x) take radians; write pi for π and sin(x)^2 for sin²x.
Your first answer to each question is saved: it counts towards your score and brings the question back for review.
Exact answers like ln(20)/ln(3) are best. Decimals need about 4 significant figures.

Coding exercise

Exercise

Taylor polynomials and the series for ln(1 + x)

Write two functions.

taylor_poly(derivs, a, x) returns the Taylor polynomial about a at x (lesson 2.7.1). The list derivs holds $f(a), f'(a), f''(a), \ldots$ , so its length minus one is the degree. math.factorial(k) gives $k!$ .
ln1p_series(x, n) returns the sum of the first n terms of $x - \frac{x^2}{2} + \frac{x^3}{3} - \cdots$ (lesson 2.7.2). If $x$ is not in $-1 < x \le 1$ , raise ValueError: the series does not converge there.

The tests use $e^x$ about 0, $x^3$ about 2 and $\sqrt{x}$ about 4, on both sides of the centre. They also check $\ln 1.05$ , the slow series at $x = 1$ , and the two values of $x$ where the series fails.

See the tests

import math


def test_exp_about_0():
    """eˣ about 0 to degree 3, at 0.5, gives 1.645833…"""
    expect_close(taylor_poly([1, 1, 1, 1], 0, 0.5), 1 + 0.5 + 0.5**2 / 2 + 0.5**3 / 6, what="taylor_poly([1, 1, 1, 1], 0, 0.5)")


def test_factorials():
    """The k! is there: [1, 1, 1, 1] about 0 at x = 1 gives 8/3, not 4"""
    expect_close(taylor_poly([1, 1, 1, 1], 0, 1), 8 / 3, what="taylor_poly([1, 1, 1, 1], 0, 1)")


def test_cubic_about_2():
    """x³ about 2, with derivatives [8, 12, 12, 6], is exact: 15.625 at 2.5"""
    expect_close(taylor_poly([8, 12, 12, 6], 2, 2.5), 2.5**3, what="taylor_poly([8, 12, 12, 6], 2, 2.5)")


def test_left_of_centre():
    """√x about 4 to degree 2, at 3.8 (left of the centre), gives 1.949375"""
    expect_close(taylor_poly([2, 1 / 4, -1 / 32], 4, 3.8), 1.949375, what="taylor_poly([2, 1/4, -1/32], 4, 3.8)")


def test_degree_0():
    """With one value, the copy is the constant f(a)"""
    expect_close(taylor_poly([5], 3, 10), 5, what="taylor_poly([5], 3, 10)")


def test_ln_two_terms():
    """Two terms at x = 0.05 give 0.04875"""
    expect_close(ln1p_series(0.05, 2), 0.04875, what="ln1p_series(0.05, 2)")


def test_ln_inside():
    """40 terms at x = −0.5 give ln 0.5 to about 13 digits"""
    expect_close(ln1p_series(-0.5, 40), math.log(0.5), what="ln1p_series(-0.5, 40)", rel=1e-12)


def test_ln_slow_end():
    """x = 1 is allowed but slow: 1,000 terms are within 0.001 of ln 2"""
    value = ln1p_series(1, 1000)
    assert value is not None and abs(value - math.log(2)) < 0.001, f"expected within 0.001 of ln 2 = {math.log(2):.6f}, got {value}"


def test_ln_beyond_1():
    """x = 1.5 raises ValueError: the series diverges there"""
    try:
        ln1p_series(1.5, 10)
    except ValueError:
        return
    raise AssertionError("expected ValueError for x = 1.5")


def test_ln_at_minus_1():
    """x = −1 raises ValueError: there the series is minus the harmonic series"""
    try:
        ln1p_series(-1, 10)
    except ValueError:
        return
    raise AssertionError("expected ValueError for x = -1")

Taylor polynomials

Standard series and the error

Coding exercise

Free further reading