Exercises on Advanced Mathematics

Duong Thanh PHAM

April 24, 2014

2

Chapter 1

SERIES

Exercise 1. Let an =n

5n+ 2.

1. Determine whether {an} is convergent.

2. Determine whether∑∞

n=1 an is convergent.

Exercise 2. Similar questions as in Exercise 1 with:

1. an =1

n(n+ 2)2. an =

1

n(n+ 5)

Exercise 3. Determine whether the series is convergent or divergent. If it is convergent, find its sum

1.

∞∑n=2

2

n2 − 12.

∞∑n=1

2

n2 + 4n+ 33.

∞∑n=1

3

n(n+ 3)

Exercise 4. Determine whether the geometric series is convergent or divergent. If it is convergent,

find its sum.

1. 3 + 2 +4

3+

8

9+ . . .

2.1

8− 1

4+

1

2− 1 + . . .

3.∞∑n=1

6(0.9)n−1

4.

∞∑n=1

10n

(−9)n−1

5.∞∑n=1

(−3)n−1

4n

6.

∞∑n=0

1

(√2)n

7.

∞∑n=0

πn

3n+1

8.

∞∑n=1

en

3n−1

Exercise 5. Determine whether the series is convergent or divergent. If it is convergent, find its sum.

1.

∞∑n=1

1

2n

2.

∞∑n=1

n+ 1

2n− 3

3.

∞∑k=2

k2

k2 − 1

4.

∞∑k=1

k(k + 2)

(k + 3)2

5.

∞∑n=1

1 + 2n

3n

6.

∞∑n=1

1 + 3n

2n

3

4 CHAPTER 1. SERIES

7.∞∑n=1

n√2

8.∞∑n=1

ln

(n2 + 1

2n2 + 1

)9.

∞∑n=1

(3

7n+

2

n

)

10.∞∑n=1

(1

en+

1

n(n+ 1)

)11.

∞∑n=1

en

n2

Exercise 6. Find the values of x for which the series converges. Find the sum of the series for those

values of x.

1.

∞∑n=1

xn

3n

2.

∞∑n=1

(x− 4)n

3.

∞∑n=0

4nxn

4.

∞∑n=0

(x+ 3)n

2n

5.

∞∑n=0

cosn x

2n

Exercise 7. Determine whether the series converges or diverges. (Hint: you may use the Comparison

Test)

1.∞∑n=1

n

2n3 + 1

2.∞∑n=1

n3

n4 + 1

3.

∞∑n=1

n+ 1

n√n

4.∞∑n=1

n− 1

n2√n

5.∞∑n=1

9n

3 + 10n

6.

∞∑n=1

4 + 3n

2n

7.∞∑n=1

cos2 n

n2 + 1

8.∞∑n=1

n2 − 1

3n4 + 1

9.

∞∑n=1

n− 1

n4n

10.∞∑n=1

1 + sinn

10n

11.∞∑n=1

√n

n− 1

12.

∞∑n=1

2 + (−1)n

n√n

13.∞∑n=1

1√n3 + 1

14.∞∑n=1

1√n2 + 1

15.

∞∑n=1

1

2n+ 3

16.∞∑n=1

1 + 4n

1 + 3n

17.∞∑n=1

n+ 4n

n+ 6n

18.

∞∑n=1

√n+ 2

2n2 + n+ 1

19.∞∑n=1

n+ 2

(n+ 1)3

20.

∞∑n=1

e1/n

n

21.

∞∑n=1

1

n!

22.

∞∑n=1

n!

nn

23.

∞∑n=1

sin1

n

Exercise 8. Determine whether the series converges or diverges. (Hint: you may use the Ratio Test

or the Root Test)

1.

∞∑n=1

n!

nn

2.∞∑n=1

3nn!

nn

3.

∞∑n=1

n!

an(a > 0)

4.∞∑n=1

2n

n+ 2n

5.

∞∑n=1

n2

(2 + 1n)

n

6.

∞∑n=1

(n2 − 5n+ 1

n2 − 4n+ 2

)n2

7.

∞∑n=1

n2

2n

8.

∞∑n=1

10n

(n+ 1)42n+1

9.∞∑n=1

n!

100n

10.

∞∑n=1

3− cosn

n2/3 − 2

11.∞∑n=1

(n2 + 1

2n2 + 1

)n

12.∞∑n=1

(1 +

1

n

)n2

13.

∞∑n=1

n

(lnn)n

14.

∞∑n=1

2 · 4 · 6 · . . . · (2n)n!

5

Exercise 9. Determine whether the series converges or diverges.

1.

∞∑n=1

ne−n

2.∞∑n=0

1

2n+ 1

3.∞∑n=1

lnn

n3

4.

∞∑n=1

1

n lnn

5.

∞∑n=1

1

n(lnn)2

6.∞∑n=1

e1/n

n2

7.

∞∑n=1

n2

en

Exercise 10. Find the values of p for which the series is convergent. (Hint: you may use the Integral

Test)

1.∞∑n=2

1

n(lnn)p2.

∞∑n=3

1

n lnn[ln(lnn)]p3.

∞∑n=1

n(1 + n2)p 4.∞∑n=1

lnn

np

Exercise 11. 1.

∞∑n=1

(−1)n

lnn

2.

∞∑n=1

(−1)n−1 3n+ 1

n(n+ 1)

3.

∞∑n=1

(−1)n 3n− 1

2n+ 1

4.

∞∑n=1

(−1)n n√n3 + 2

5.

∞∑n=1

(−1)n n

10n

6.

∞∑n=1

(−1)n n

lnn

7.

∞∑n=1

cosnπ

n3/4

8.

∞∑n=1

(−1)n sin πn

9.

∞∑n=1

(−1)n cos πn

Exercise 12. Determine whether the series is absolutely convergent, conditionally convergent, or

divergent.

1.∞∑n=1

(−1)n

n!

2.∞∑n=1

(−1)n−1 2n

n4

3.∞∑n=1

(−1)n+1

4√n

4.∞∑n=1

(−1)n

n4

5.∞∑n=1

n!

100n

6.∞∑n=1

(−1)n n√n3 + 2

Exercise 13. Find the radius of convergence and interval of convergence of the series.

1.

∞∑n=1

xn√n

2.

∞∑n=1

(−1)nxn

n+ 1

3.

∞∑n=1

(−1)n−1xn

n3

4.

∞∑n=1

√nxn

5.xn

n!

6. nnxn

7. (−1)nn2xn

2n

8.10nxn

n3

9.(−2)nxn

4√n

10.xn

5nn5

11.

∞∑n=2

(−1)n xn

4n lnn

12.

∞∑n=0

(−1)n x2n

(2n)!

13.

∞∑n=0

(x− 2)n

nn

14.

∞∑n=0

(−1)n (x− 3)n

2n+ 1

15.

∞∑n=1

3n(x+ 4)n√n

16.

∞∑n=1

n

4n(x+ 1)n

17.

∞∑n=1

(x− 2)n

nn

6 CHAPTER 1. SERIES

Exercise 14. Find a power series representation for ln(1− x) and its radius of convergence.

Exercise 15. Find a power series representation for f(x) =x2

1− x2

Exercise 16. Find the Maclaurin series for cosx

Exercise 17. Find the Maclaurin series for x cosx

Exercise 18. Find the Maclaurin series for

1.1

1− x2. xex

3. ln(1 + x)

4. cos 3x

5. ex + e3x

6. sin2 x

7. cosx2

8. ln(1 + x2)

Exercise 19. Use the Maclaurin series for ex to calculate e−0.2 correct to five decimal places.

Exercise 20. Find the Maclaurin series of tanx.

Chapter 2

Vector and geometry of space

Exercises in Calculus book by James Stewart

• Exercises 1–7; 17; 18; 19 (page 792)

• Exercises 29–32; 38; 46 (page 793)

• Exercises 1–15; 19–38; 43–45; 49–54; 64–66 (page 802–803)

• Exercises 69–73 (page 803–804)

• Exercises 1–20 (page 810–811)

7

8 CHAPTER 2. VECTOR AND GEOMETRY OF SPACE

Chapter 3

Partial Derivatives

Exercise 21. Find the limit, if it exists, or show that the limit does not exist

1. lim(x,y)→(1,2)

(5x3 − x2y2)

2. lim(x,y)→(1,−1)

e−xy cos(x+ y)

3. lim(x,y)→(2,1)

4− xyx2 + 3y2

4. lim(x,y)→(0,0)

y4

x4 + 3y4

5. lim(x,y)→(0,0)

x2 + sin2 y

2x2 + y2

6. lim(x,y)→(0,0)

xy√x2 + y2

7. lim(x,y)→(0,0)

x4 − y4

x2 + y2

8. lim(x,y)→(0,0)

x2yey

x4 + 4y2

9. lim(x,y)→(0,0)

x2 sin2 y

x2 + 2y2

10. lim(x,y)→(0,0)

xy4

x2 + y8

Exercise 22. Determine if the limit exists

lim(x,y)→(0,0)

xy

x2 + y2

Exercise 23. Determine if the limit exists

lim(x,y)→(0,0)

xy4

x2 + y4

Exercise 24. Determine if the limit exists

lim(x,y)→(0,0)

3x2y

x2 + y2

Exercise 25. Use polar coordinates to find the following limits

1. lim(x,y)→(0,0)

x3 + y3

x2 + y2

2. lim(x,y)→(0,0)

(x2 + y2) ln (x2 + y2)

3. lim(x,y)→(0,0)

e−x2−y2 − 1

x2 + y2

Exercise 26. Determine the set of points at which the function is continuous.

9

10 CHAPTER 3. PARTIAL DERIVATIVES

1. F (x, y) =sin(xy)

ex − y2

2. F (x, y) =x− y

1 + x2 + y2

3. F (x, y) = ex2y +

√x+ y2

4. F (x, y) = ln(x2 + y2 − 4)

5. F (x, y) =√x+ y + z

Exercise 27. Determine if f is continuous at (0, 0)

f(x, y) =

x2−y2x2+y2

if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0)

Exercise 28. Determine if g is continuous at (0, 0)

g(x, y) =

3x2yx2+y2

if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0)

Exercise 29. Determine if g is continuous at (0, 0)

g(x, y) =

x2y3

2x2+y2if (x, y) 6= (0, 0)

1 if (x, y) = (0, 0)

Exercise 30. Determine if h is continuous at (0, 0)

h(x, y) =

sinxyxy if (x, y) 6= (0, 0)

1 if (x, y) = (0, 0)

Exercise 31. Find the first partial derivatives of the function.

1. f(x, y) = y5 − 3xy

2. f(x, y) = x4y3 + 8x2y

3. f(x, t) = e−t cosπx

4. f(x, t) =√x ln t

5. f(x, y) = x ln(x2 + y2)

6. u =√x21 + x22 + . . .+ x2n

7. u = sin(x1 + 2x2 + . . .+ nxn)

8. u = xy/z

Exercise 32. Find the indicated partial derivatives

1. f(x, y) = ln(x+√x2 + y2; fx(3, 4)

2. f(x, y) = arctan(y/x); fx(2, 3)

3. f(x, y, z) = yx+y+z ; fy(2, 1,−1)

4. f(x, y, z) =√

sin2 x+ sin2 y + sin2 z; fz(0, 0, π/4)

Exercise 33. Find all the second partial derivatives

11

1. f(x, y) = x3y5 + 2x4y

2. f(x, y) = sin2(mx+ ny)

3. w =√u2 + v2

4. v = exey

Exercise 34. Use definition to find the partial derivatives fx(x, y) and fy(x, y)

1. f(x, y) = xy2 − x3y 2. f(x, y) =x

x+ y2

Exercise 35. Determine whether each of the following functions is a solution of the Laplace’s equation

uxx + uyy = 0.

1. u = x2 + y2

2. u = x2 − y2

3. u = x3 + 3xy2

4. u = ln√x2 + y2

Exercise 36. Verify that the function u = 1/√x2 + y2 + z2 is a solution of the three-dimensional

Laplace equation uxx + uyy + uzz = 0.

Exercise 37. Let

f(x, y) =

x3y − xy3

x2 + y2if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0).

1. Prove that f is continuous in R2.

2. Find fx(x, y) and fy(x, y) when (x, y) 6= (0, 0).

3. Find fx(0, 0) and fy(0, 0).

4. Show that fxy(0, 0) = −1 and fyx(0, 0) = 1.

Exercise 38. Find the equation of the tangent plane to the given surface at the specified point.

1. z = 4x2 − y2 + 2y, (−1, 2, 4)

2. z = 3(x− 1)2 + 2(y + 3)2 + 7, (2,−2, 12)

3. z = y lnx, (1, 4, 0)

Exercise 39. Explain why the function is differentiable at the given point. Then find the linearization

L(x, y) of the function at that point.

1. f(x, y) = x√y, (1, 4)

2. f(x, y) = x3y4, (1, 1)

3. f(x, y) = xx+y (2, 1)

Exercise 40. Find the linear approximation of the function

f(x, y) =√20− x2 − 7y2

at (2, 1) and use it to approximate f(1.95, 1.08).

12 CHAPTER 3. PARTIAL DERIVATIVES

Exercise 41. Let

f(x, y) =

xy

x2+y2if (x, y) 6= (0, 0)

0 if (x, y) = (0, 0).

1. Show that that fx(0, 0) and fy(0, 0) both exist but f is not differentiable at (0, 0).

2. Prove that fx and fy are not continuous at (0, 0).

Exercise 42. Use the chain rule to find dz/dt

1. z = x2 + y2 + xy, x = sin t, y = et

2. z = cos(x+ 4y), x = 5t4, y = 1/t

3. z =√

1 + x2 + y2, x = ln t, y = cos t

4. z = ln√x2 + y2 + z2, x = sin t, y = cos t, z = tan t.

Exercise 43. Use the chain rule to find ∂z/∂s and ∂z/∂t.

1. z = x2y2, x = s cos t, y = s sin t

2. z = sin θ cosϕ, θ = st2, ϕ = s2t

3. z = er cos θ, r = st, θ =√s2 + t2.

Exercise 44. If z = f(x, y), where f is differentiable, and

x = g(t) y = h(t)

g(3) = 2 h(3) = 7

g′(3) = 5 h′(3) = −4fx(2, 7) = 6 fy(2, 7) = −8,

find dz/dt when t = 3.