4.5-4.6 Maclaurin series and standard series Quiz: Pearson Edexcel Further Maths, Unit 4

The 20 questions

1. What is the Maclaurin series for e^x?

   • 1 + x^2/2! + x^4/4! + ...
   • 1 + x + x^2/2! + x^3/3! + ...
   • 1 - x + x^2/2! - x^3/3! + ...
   • 1 + x + x^2 + x^3 + ...

2. What is the first non-zero term of the Maclaurin series of sin x after the x term?

   • -x^3/3!
   • +x^3/3!
   • -x^2/2!
   • -x/1!

3. What is the Maclaurin series of cos x up to the x^4 term?

   • 1 + x^2/2! + x^4/4!
   • 1 - x + x^2/2!
   • x - x^3/3!
   • 1 - x^2/2! + x^4/4!

4. For which values of x is the Maclaurin series of ln(1 + x) valid?

   • all real x
   • -1 < x <= 1
   • -1 <= x < 1
   • |x| > 1

5. For which values of x is the binomial series for (1 + x)^n valid when n is not a positive integer?

   • all real x
   • |x| < 1
   • x > 0 only
   • |x| > 1

6. What is the general term of the Maclaurin series of e^x?

   • (-1)^n x^n
   • x^n/n
   • n x^n
   • x^n/n!

7. Which expression gives the coefficient of x^n in the Maclaurin series of f(x)?

   • f^(n)(0) / n!
   • f^(n)(0) x n!
   • f'(0) / n
   • f^(n)(0) / n

8. What is the Maclaurin series of e^(2x) up to the x^2 term?

   • 1 + 2x + x^2
   • 1 + 2x + 2x^2
   • 1 + x + 2x^2
   • 1 + 2x + 4x^2

9. What is the Maclaurin series of cos(2x) up to the x^4 term?

   • 1 - 2x^2 + x^4
   • 1 - 2x^2 + (2/3)x^4
   • 1 - x^2 + (2/3)x^4
   • 1 - 2x^2 + (4/3)x^4

10. Using the first two non-zero terms of the Maclaurin series of sin x, what is the approximate value of sin(0.1)?

    • 0.1
    • 0.09
    • 0.10017
    • 0.09983

11. What is the coefficient of x^3 in the Maclaurin series of ln(1 + x)?

    • -1/3
    • 1/3
    • 1/6
    • 1/2

12. What is the Maclaurin series of (1 + x)^(1/2) up to the x^2 term?

    • 1 + x/2 + x^2/8
    • 1 + x/2 - x^2/8
    • 1 + x - x^2/8
    • 1 - x/2 - x^2/8

13. What is the coefficient of x^3 in the Maclaurin series of (1 + x)^(-1)?

    • 1
    • 3
    • -3
    • -1

14. What is the Maclaurin series of e^(x^2) up to the x^4 term?

    • 1 + x^2 + x^4/4
    • 1 + x^2 + x^4
    • 1 + x^2 + x^4/2
    • 1 + 2x^2 + x^4/2

15. What is the value of the limit of (1 - cos x)/x^2 as x tends to 0, using Maclaurin series?

    • 2
    • 1/2
    • 1
    • 0

16. What is the Maclaurin series of x cos x up to the x^3 term?

    • 1 - x^2/2
    • x - x^3/2
    • x - x^3/6
    • x + x^3/2

17. What is the value of the limit of (e^x - 1 - x)/x^2 as x tends to 0?

    • -1/2
    • 0
    • 1/2
    • 1

18. Using the first three non-zero terms of the Maclaurin series of ln(1 + x), what is the approximation to ln(1.5)?

    • 7/12
    • 3/8
    • 1/2
    • 5/12

19. What is the Maclaurin series of sin(x)/x up to the x^2 term?

    • 1 + x^2/6
    • x - x^3/6
    • 1 - x^2/6
    • 1 - x^2/2

20. What is the Maclaurin series of e^(-x) sin x up to the x^2 term?

    • x - x^2/2
    • x - x^2
    • 1 - x
    • x + x^2

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