THE CHINESE UNIVERSITY OF HONG KONGDepartment of Mathematics

MATH1020 General Mathematics (Fall 2017)Homework 4

Due Date: November 17, 2017

Name: Student No.:

Class: Final Result:

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Signature Date

For instructor’s use only

Graded problems

Score

Total

2

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Instructions for Homework 4.

• Please attempt to solve all the problems. Your solutions of problems 1 − 10 are tobe submitted, a part of which will be graded.

• Problems 11 − 14 are strongly recommended for you to study, though you are notrequired to submit their solutions.

3

1. Find all coefficients:

(a)5x+ 1

(x− 1)(x+ 2)=

A

x− 1+

B

x+ 2;

(b)2x2 + 12x+ 6

(x+ 1)(x+ 2)(x− 3)=

A

x+ 1+

B

x+ 2+

C

x− 3;

(c)5x2 − 3x+ 1

(x− 1)2(x+ 2)=

A

(x− 1)2+

B

x− 1+

C

x+ 2.

4

5

2. Find if the series converges or diverges. If convergent then find its sum.

(a)∞∑k=1

6

(2k − 1)(2k + 1).

(b)∞∑k=1

(sin2 1

k− sin2 1

k + 2

).

(c)∞∑k=1

ln

(k

k + 1

).

(d)∞∑k=1

(7 · 3k+1

22k+3+ e−k

)

6

7

3. Determine if the series converges or diverges.

(a)∞∑k=1

ek

3k−1 ; (b)∞∑k=1

k(k + 2)

(k + 3)2;

(c)∞∑k=1

cos

(kπ

2

); (d)

∞∑k=1

2k + 1

k2(k + 1)2;

(e)∞∑k=1

2k+1 − 3

3k + (−1)k; (f)

∞∑k=1

k−k =∞∑k=1

1

kk.

8

9

Definition 1 The number of combinations of r objects taken from a group of n distinctobjects is denoted by nCr and is given by:

nCr =

(n

r

)=

n!

(n− r)! r!.

Theorem 1 The binomial expansion of (a+ b)n for any positive integer n is:

(a+ b)n = nC0anb0 + nC1a

n−1b1 + nC2an−2b2 + · · ·+ nCna

0bn =n∑

r=0

nCran−rbr.

4. Use the Binomial Theorem to expand and simplify:

(a)

(x2

2− 2

x

)4

;

(b) (1− i)6, where i is defined to be√−1 (i is called the imaginary unit);

(c)

(a− 1

a

)6

, where a is a positive constant.

10

11

Theorem 2 Binomial Series

If |x| < 1, then for any real number r.

(1 + x)r =∞∑k=0

(rk

)xk,

where(r0

)= 1, k = 0, and

(rk

)=r(r − 1)(r − 2) · · · (r − k + 1)

k!, k ≥ 1.

5. Answer the following questions:

(a) Find the middle term of

(1− x2

2

)14

;

(b) Find the coefficient of x19 in (2x3 − 3x)9.

(c) Use the binomial theorem to determine, correct to 4 decimal places

i. (1.003)8;

ii. (0.98)7.

12

13

6. Answer the following questions using the binomial series:

(a) Expand1

(2 + x)3in ascending powers of x as far as the term in x3. State in

each case the limits of x for which the series is valid.

(b) Expand (2+3x)−6 to three terms. For what values of x is the expansion valid?

14

15

7. When x is very small show that:

(a)1

(1− x)2√

1− x≈ 1 +

5

2x;

(b)1− 2x

(1− 3x)2≈ 1 + 4x.

16

8. A magnetic pole, distance x from the plane of a coil of radius r, and on the axis ofthe coil, is subject to a force F when a current flows in the coil. The force is givenby

F =kx√

(r2 + x2)5,

where k is a constant. Use the binomial theorem to show that when x is smallcompared to r, then

F ≈ kx

r5− 5kx3

2r7.

17

Basic Taylor Series

1

1− x= 1 + x+ x2 + · · ·+ xn + · · · , −1 < x < 1.

1

1 + x= 1− x+ x2 + · · ·+ (−1)nxn + · · · , −1 < x < 1.

ln(1− x) = −x− 1

2x2 − 1

3x3 − · · · − 1

nxn − · · · , −1 < x < 1.

ln(1 + x) = x− 1

2x2 +

1

3x3 − · · ·+ (−1)n−1

nxn + · · · , −1 < x < 1.

ex = 1 + x+1

2!x2 + · · ·+ 1

n!xn + · · · , −∞ < x <∞.

e−x = 1− x+1

2!x2 − · · ·+ (−1)n

n!xn + · · · , −∞ < x <∞.

9. Find the general nth term as follows:

(a)1

1− x+

1

1 + x;

(b) x ln(1− x);

(c) x2 ln(1 + x);

(d) e2x;

(e) e−(x+1).

18

19

10. Consider the power series

sin x =∞∑n=0

(−1)nx2n+1

(2n+ 1)!= x− x3

6+

x5

120− . . . for all x ∈ R,

cos x =∞∑n=0

(−1)nx2n

(2n)!= 1− x2

2+x4

24− x6

720+ . . . for all x ∈ R,

Verify that the power series of the left hand side and right hand side of each of thefollowing trignometric identies are equal up to the x6 term.

(a) sin 2x = 2 sin x cos x;

(b) sin2 x+ cos2 x = 1.

20

21

11. (Optional) Let R be the radius of convergence of a power series p(x) =∞∑n=0

cnxn. It

is a fact that p(x) is continuous on (−R,R). In particular, if R 6= 0, we have

limx→0

p(x) = p(0) = c0.

Given power series p(x), q(x) with non-zero radii of convergence, the limit limx→0

p(x)

q(x)can be computed by comparing the lowest order non-zero terms of p(x) and q(x).For instance,

limx→0

x2 + 3x3 + x4 + . . .

3x2 + x3 − 2x4 + . . .= lim

x→0

1 + 3x+ x2 + . . .

3 + x− 2x2 + . . .=

1

3;

limx→0

x2 + 2x3 + 3x4 + . . .

3x+ 2x2 + x3 + . . .= lim

x→0

x+ 2x2 + 3x3 + . . .

3 + 2x+ x2 + . . .=

0

3= 0;

limx→0

x+ 2x2 + 3x3 + . . .

3x2 + 2x3 + x4 + . . .= lim

x→0

1 + 2x+ 3x2 + . . .

3x+ 2x2 + x3 + . . .= +∞;

limx→0

x+ 2x2 + 3x3 + . . .

−3x2 + 2x3 + x4 + . . .= lim

x→0

1 + 2x+ 3x2 + . . .

−3x+ 2x2 + x3 + . . .= −∞.

Use the method above and the power series

sin x =∞∑n=0

(−1)nx2n+1

(2n+ 1)!= x− x3

6+ . . . for all x ∈ R,

ex =∞∑n=0

xn

n!= 1 + x+

x2

2+x3

6. . . for all x ∈ R,

√1 + x =

∞∑n=0

(12

n

)xn = 1 +

x

2− x2

8+x3

16+ . . . for all |x| < 1,

evalulate the following limits

(a) limx→0

sinx− xx(ex − 1)

;

(b) limx→0

x3

ex − sin x−√

1 + x2;

(c) limx→0

√1 + 2x+

√1− 2x√

1 + 3x+√

1− 3x.

22

12. (Optional) Use the Binomial Theorem to prove the following identity:

(a)n∑

i=0

(n

i

)2

=

(2n

n

), n ≥ 0.

(b)n∑

k=0

(−1)k(n

k

)= 0, n ≥ 1.

(c) Let n be a non-negative integer.

n∑k=0

2k

(n

k

)= 3n.

13. (Optional) In the theory related to the dispersion of light, the expression

1 +A

1− λ20/λ2

arises. Answer the following questions:

(a) Let x = λ20/λ2 and find the first five terms of expansion of (1− x)−1.

(b) Write the original expression in expanded form using the result of (a).

14. (Optional) Answer the following questions separately:

(a) If term a1 is given along with a rule to find term an+1 from term a1, the sequenceis said to be defined recursively. If a1 = 2 and an+1 = (n+ 1)an, find the firstfive terms of the sequence.

(b) A sequence is defined recursively by x1 =N

2, xn+1 =

1

2

(xn +

N

xn

). With

N = 10, find x6 and compare the value with√

10. It can be seen that√N can

be approximated using this recursion sequence.