1.5.4 Practice set: series

Sequences

Sequences (lesson 1.5.1)

0 of 7 correct

1.
If $a_n = 5n + 2$ , find $a_{12}$
2.
If $a_1 = 4$ and $a_{n+1} = 2a_n - 3$ , find $a_4$
3.
Find the 8th term of $3, 6, 12, \ldots$
4.
An arithmetic sequence has $a_1 = 7$ and $a_{10} = 43$ . Find the common difference $d$
5.
Find the limit of $a_n = 3 + 2\left(\tfrac{1}{2}\right)^n$
6.
$\frac{1}{n}$ has limit 0. From which term on is it closer than 0.05 to 0?
7.
A sequence starts at $a_1 = 100$ and grows by 10% a step. Find $a_3$

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.

Finite series

Arithmetic and geometric series (lesson 1.5.2)

0 of 7 correct

1.
A sequence has partial sums $S_6 = 51$ and $S_5 = 40$ . Find $a_6$
2.
Find the sum of the whole numbers from 1 to 200
3.
Find $5 + 8 + 11 + \cdots + 50$
4.
Find $2 + 6 + 18 + \cdots$ to 6 terms
5.
Find $\displaystyle\sum_{k=2}^{6} 3^k$
6.
Find $4 - 8 + 16 - 32 + 64$
7.
Find $\displaystyle\sum_{k=1}^{12} 5$

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.

Infinite series

Infinite series (lesson 1.5.3)

0 of 7 correct

1.
Find $1 + \tfrac{1}{4} + \tfrac{1}{16} + \cdots$
2.
Find $\displaystyle\sum_{k=1}^{\infty} 5\,(0.2)^k$
3.
Write $0.454545\ldots$ as a fraction
4.
An infinite geometric series has first term 3 and sum 2. Find its ratio $r$
5.
$1 + \tfrac{1}{3} + \tfrac{1}{9} + \cdots$ adds to $\tfrac{3}{2}$ . How far is its partial sum $S_5$ from that?
6.
How many terms of $\tfrac{1}{2} + \tfrac{1}{4} + \cdots$ bring the partial sum within 0.0001 of 1?
7.
The harmonic sum $1 + \tfrac{1}{2} + \tfrac{1}{3} + \cdots$ is 7.485 after 1,000 terms, and each tenfold increase in $n$ adds about 2.303. Estimate it after 10,000 terms

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

Geometric sums, finite and infinite

Write three functions.

geometric_sum(a, r, n) returns $a + ar + \cdots + ar^{n-1}$ , the sum of $n$ terms. Handle $r = 1$ , where the formula divides by zero, and $n = 0$ , an empty sum.
geometric_sum_infinite(a, r) returns $\frac{a}{1 - r}$ . Raise ValueError when $|r| \ge 1$ , because the series diverges.
terms_needed(a, r, tol) returns the smallest number of terms $n$ whose partial sum is within tol of the infinite sum. Check the ratio first and raise ValueError when $|r| \ge 1$ : there the gap never shrinks, so a loop would run for ever.

The tests check terms_needed against partial sums added up directly, including a negative ratio and a negative first term.

See the tests

def test_doubling():
    """5 + 10 + 20 + … to 10 terms is 5115"""
    expect_close(geometric_sum(5, 2, 10), 5115, what="geometric_sum(5, 2, 10)")


def test_matches_direct_sum():
    """The formula matches a direct sum: 1000 at ratio 1.05, 7 terms"""
    expect_close(geometric_sum(1000, 1.05, 7), sum(1000 * 1.05**k for k in range(7)), what="geometric_sum(1000, 1.05, 7)")


def test_ratio_one():
    """With ratio 1, every term is a: 7 terms of 3 make 21"""
    expect_close(geometric_sum(3, 1, 7), 21, what="geometric_sum(3, 1, 7)")


def test_no_terms():
    """Zero terms make an empty sum, 0"""
    expect_close(geometric_sum(4, 0.5, 0), 0, what="geometric_sum(4, 0.5, 0)")


def test_infinite():
    """1/2 + 1/4 + … adds to 1"""
    expect_close(geometric_sum_infinite(0.5, 0.5), 1.0, what="geometric_sum_infinite(0.5, 0.5)")


def test_infinite_negative_ratio():
    """5 − 5/2 + 5/4 − … adds to 10/3"""
    expect_close(geometric_sum_infinite(5, -0.5), 10 / 3, what="geometric_sum_infinite(5, -0.5)")


def test_diverges():
    """A ratio of 2 or −1 diverges: raise ValueError"""
    for r in (2, -1):
        try:
            geometric_sum_infinite(1, r)
        except ValueError:
            continue
        raise AssertionError(f"expected ValueError for r = {r}")


def _gap(a, r, n):
    """How far the partial sum of n terms, added up directly, is from a / (1 - r)."""
    return abs(a / (1 - r) - sum(a * r**k for k in range(n)))


def test_terms_needed_halves():
    """1/2 + 1/4 + … needs 20 terms to come within 0.000001 of 1"""
    expect_equal(terms_needed(0.5, 0.5, 1e-6), 20, what="terms_needed(0.5, 0.5, 1e-6)")


def test_terms_needed_slow_ratio():
    """1 + 0.9 + 0.81 + … needs 66 terms to come within 0.01 of 10"""
    expect_equal(terms_needed(1, 0.9, 0.01), 66, what="terms_needed(1, 0.9, 0.01)")


def test_terms_needed_is_smallest():
    """For a negative ratio and a negative first term, n terms are close enough and n − 1 are not"""
    for a, r, tol in [(1, -0.5, 0.001), (-2, 0.6, 0.001), (3, 0.25, 1e-5)]:
        n = terms_needed(a, r, tol)
        assert isinstance(n, int) and n >= 1, f"terms_needed({a}, {r}, {tol}) should be a whole number of terms, got {n!r}"
        assert _gap(a, r, n) < tol, f"terms_needed({a}, {r}, {tol}) returned {n}, but {n} terms still miss by {_gap(a, r, n):.3g}"
        assert _gap(a, r, n - 1) >= tol, f"terms_needed({a}, {r}, {tol}) returned {n}, but {n - 1} terms are already close enough"


def test_terms_needed_diverges():
    """A ratio of 1.5 has no sum to approach: raise ValueError"""
    try:
        terms_needed(1, 1.5, 0.01)
    except ValueError:
        return
    raise AssertionError("expected ValueError for r = 1.5")

Sequences

Finite series

Infinite series

Coding exercise

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