AP Calculus Convergence of Series Student Handout

Convergence of Series

Student Handout

2016-2017 EDITION

Use the following link or scan the QR code to complete the evaluation for the Study Session https://www.surveymonkey.com/r/S_SSS

Convergence of Series

Students should be able to:

 Recognize various types of numerical series and efficiently apply the appropriate test.  Understand that a series may be absolutely convergent, conditionally convergent or divergent and utilize proper techniques to decide.  Determine the sum of an infinite geometric series and be able to use that sum to create a power series and determine its interval of convergence.  Understand that an infinite series of numbers converges to a real number S (or has sum S), if and only if the limit of its sequence of partial sums exists and equals to S.  Use the methods the nth term test, the comparison test, the limit comparison test, the geometric series test, p-series test, the integral test, the ratio test and the alternating series test for determining whether the series of numbers converges or diverges.  Use the ratio test to determine radius or open interval of convergence of power series.  Use the other tests to check convergence at the endpoints.  Use the alternating series error bound, if an alternating series converges, to estimate how close a partial sum is to the value of the infinite series.

Multiple Choice

1. (calculator not allowed)  n n  2 n What is the interval of convergence for the power series 1 n x  4 ? n0 n  3

(A) 3  x  3 (B) 3  x  3 (C) 1  x  7 (D) 1  x  7

2. (calculator not allowed) Which of the following series are convergent? 1 1 1 I. 1    . . .   . . . 2 2 32 n 2 1 1 1 II. 1   . . .   . . . 2 3 n 1 1 1 n1 III. 1   . . .   . . . 3 32 3n1

(A) I only (B) III only (C) I and III only (D) II and III only (E) I, II, and I

3. (calculator not allowed) The Taylor series for a function f about x  0 converges to f for 1  x  1. The nth-degree n k k x Taylor Polynomial for f about x  0 is given by Pn x 1 2 . Of the following, k1 k  k 1 which is the smallest number M for which the alternating series error bound guarantees that

f 1  P4 1  M ? 1 1 (A)  5! 31 1 1 (B)  4! 21 1 (C) 31 1 (D) 21

4. (calculator not allowed)  1n Which of the following statements about the series  is true? n1 1 n (A) The series converges absolutely. (B) The series converges conditionally. (C) The series converges but neither conditionally nor absolutely. (D) The series diverges.

5. (calculator not allowed)

Which of the following series can be used with the limit comparison test to determine whether the  n series  3 converges or diverges? n1 n 1  1 (A)  n1 n  1 (B)  2 n1 n  1 (C)  3 n1 n

 n3 1 (D)  2 n1 n

6. (calculator not allowed)

Which of the following series are conditionally convergent? n n  1  1 n  1 I. II.  III.  4  3 2  n1 n n1 n n1 n

(A) III only (B) I and II (C) II and III (D) I, II and III

7. (calculator not allowed)

Which of the following series is absolutely convergent?

 n1 1 (A)  1 n 1 2n  n1 1 (B)  1 n 1 n  n1 n (C)  1 n 1 n 1  n n1  1 (D)  1   n 1  2

8. (calculator not allowed) Which of the following series converge?  n I.  n 1 n  2  cosn II.  n 1 n  1 III.  n 1 n

(A) None (B) II only (C) III only (D) I and II only (E) I and III only

9. (calculator not allowed) b dx If lim is finite, then which of the following must be true? b   1 x p

 1 (A) converges  p n 1 n  1 (B) diverges  p n 1 n  1 (C) converges  p2 n 1 n  1 (D) converges  p 1 n 1 n  1 (E) diverges  p 1 n 1 n

10. (calculator not allowed)  2n 1 What is the value of ?  n n 1 3

(A) 1 (B) 2 (C) 4 (D) 6 (E) The series diverges

11. (calculator not allowed)  n What are all values of p for which the infinite series converges?  p n 1 n 1

(A) p  0 (B) p1 (C) p1 (D) p  2 (E) p  2

12. (calculator not allowed) Which of the following series diverge?

 n  sin 2  I.    n 0     1 II.  3 n 1 n   en  III.     n  n 1 e 1

(A) III only (B) I and II only (C) I and III only (D) II and III only (E) I, II, and III

13. (calculator not allowed)  en Consider the series  . If the ratio test is applied to the series, which of the following inequalities n1 n! results, implying that the series converges?

e (A) lim 1 n n! n! (B) lim 1 n e n 1 (C) lim 1 n e e (D) lim 1 n n 1 e (E) lim 1 n (1)!n 

14. (calculator not allowed) Which of the following series converges for all real numbers x ?

 xn (A)  n1 n

 xn (B)  2 n1 n

 xn (C)  n1 n

 exnn (D)  n1 n!

 nx! n (E)  n n1 e

15. (calculator not allowed)  2 n What are all values of x for which the series  converges?  2 n1 x  1 (A) 11x  (B) x  1 only (C) x  1 only (D) x 1 and x  1 only (E) x 1 and x  1

16. (calculator not allowed)  1  p n For what values of p will both series and   converge?  2 p    n 1 n n 1  2  (A) 2  p  2 1 1 (B)   p  2 2 1 (C)  p  2 2 1 (D) p  and p  2 2 (E) There are no such values of p.

17. (calculator not allowed)

For x  1, the continuous function g is decreasing and positive. A portion of the graph of g is   shown above. For n  1, the nth term of the series a is defined by a  g(n). If g(x)dx  n n  n 1 1 converges to 8, which of the following could be true?

 (A)  an  6 n 1  (B)  an  8 n 1  (C)  an  10 n 1  (D) diverges  an n 1

18. (calculator allowed)  The power series n converges conditionally at . Which of the following  bn x  2 x  1 n 1 statements about the convergence of the series at x  6is true?

(A) The series converges conditionally at x  6

(B) The series diverges at x 6

(D) The convergence at x 6 cannot be determined using the given information

19. (calculator not allowed) n   x  The Maclaurin series for the function f is given by f (x)   . What is the value of n0  4  f (3)?

(A) 3 3 (B)  7 4 (C) 7 13 (D) 16 (E) 4

20. (calculator allowed) 

If the series  an converges and an  0 for all n, which of the following must be true? n1 a (A) lim n1  0 n an

(B) an  1 for all n 

(D)  nan diverges. n1  a (E)  n converges. n1 n

21. (calculator not allowed)  n The infinite series  ak has nth partial sum Sn  for n  1. What is the sum of the k1 1 4n 

series  ak ? k1

(A) The series diverges. 1 (B) 4 1 (C)  4 1 (D)  3

22. (calculator not allowed)

 2  4  6  8 Find the sum of the series 1     2! 4! 6! 8!

(A) -1 (B) 0 (C) 1 (D) The series diverges

Free Response

23. (calculator not allowed)  1n  1 (c) Give a value of p such that converges, but diverges. Give reasons  p  2 p n1 n n 1 n why your value of p is correct.

 1  1 (d) Give a value of p such that diverges, but converges. Give reasons why  p  2 p n1 n n 1 n your value of p is correct.

24. (calculator not allowed)

The function f is defined by the power series

 x  2 n x  2 x  22 x  23 x  2n f (x)   n  1  2  3  n  3 n 1 32 3  3 3  4 3 n 1 n1

for all real numbers x for which the series converges.

(a) Determine the interval of convergence of the power series for f. Show the work that leads to your answer.

(c) Use the first three nonzero terms of the power series for f to approximate f (1) . Use the alternating series error bound to show that this approximation differs from f (1) by less 1 than . 100

25. (calculator not allowed)

The function g has derivatives of all orders, and the Maclaurin series for g is  21n 3 5 n xxxx 1 ... n0 23357n 

(a) Using the ratio test, determine the interval of convergence of the Maclaurin series for g .

1 (b) The Maclaurin series for g evaluated at x  is an alternating series whose terms 2 1 decrease in absolute value to 0. The approximation for gusing the first two nonzero 2 17 1 terms of this series is . Show that this approximation differs from gby less than 120 2 1 . 200

Scoring rubric for question 25:

Student Samples for Question 25: Sample A

Sample B

Sample C

Test Name The series…. will converge if or will diverge if Comments th  This test can not or fails to This test is for n - term test lim an  0 a be used for n divergence  n exist n1 convergence only, the converse is false. Geometric Series  If convergent, n1 r 1 r 1 a r a  n (or if 1  r  1) (or if r  1 or r  1) Sum  n1 1 r

P-Series  1 p  1 0  p  1 When p  1,

 p the series is n1 n called harmonic series Direct  The series must 0  an  bn and 0  bn  an and Comparison a all have positive  n  n1  terms. bn diverges bn converges n1 n1 Limit  The series must an  0 , bn  0 an  0 , bn  0 Comparison a all have positive  n a a n1 lim n  L  0 lim n  L  0 terms. n n bn bn  

bn converges bn diverges n1 n1 Ratio Test an1  If lim  1 an1 an1 n an an lim 1 lim 1  n n n1 an an The ratio test fails (can not be used) Alternating  i. the terms Error Bound n1 Series (1) a or alternate in sign  n S  Sn  an1 n1 The error of  ii. decrease in This test cannot be (1)n a or estimating the  n absolute value used for divergence. sum by using n1  an1  an the first n terms n1 is less than the (1) an n1 iii. have a limit of first omitted

(the exponent zero liman  0 term must generate n alternating signs) Integral Test    f must be a and f (x)dx f (x)dx  n   positive, n1 1 1 converges diverges continuous and an  f (n)  0 (They do not decreasing converge to the same number)



When using the integral test, we compare the convergence of a series  an and an integral n1   f (x)dx . If one diverges, so does the other. If one converges, so does the other. However, they 1 don’t converge to the same number. The upper and lower bounds for the series is given by    f (x)dx  a  a  f (x)dx   n 1  1 n1 1

Absolute and Conditional Convergence:

Absolute Convergence and Rearrangement of Terms

Definition: is absolutely convergent if converges. If a series is absolutely  a n  a n convergent, then its terms can be rearranged in any order without changing the sum of the series. Example:  (1) n 1 1 1 1 1 3 1...  n       absolutely converges to but can be rearranged as: n0 3 3 9 27 81 243 4 11 11 1 9363 a 1...      using S  1 9 81 3 27 243 8 8 8 4 1  r 11111 22 2 3 or 1    ...   ... 3 9 27 81 243 327243 4

Conditional Convergence and Rearrangement of Terms

Definition: is conditionally convergent if converges but diverges.  a n  a n  a n If a series is conditionally convergent, then its terms can be rearranged to give a different sum. Example:  (1) n1  (converges by AST but absolute value diverges) n0 n  (1) n1 1 1 1  1...ln2     n0 n 234 (from Taylor Polynomial for f ()xx ln(1 ) which converges on (-1, 1]) However: 111 1111111     1111   1  1 ... 1        ...       ... 234 24368510     246810    11111 1 1...(ln2)   22345 2

MacLaurin series to be memorized:

xx23 x xnn x exx 1    ...   ...  ,  is the interval of convergence. 1! 2! 3!nn !n0 !

xxx357 x 21nn x 21 sinxx    ... ( 1)nn  ...  (  1) ;  x 3! 5! 7! (2nn 1)!n0 (2 1)!

xxx246 x 2nn x 2 cosxx 1    ...  ( 1)nn  ...  (  1) ;  2! 4! 6! (2nn )!n0 (2 )!

Nice to know: geometric power series:

1  1x xx23   ...  xnn  ...  x ;  1 x  1 1 x n0