University:
Washington State University
Course:
MATH 172 | Calculus II
Academic year:

2024
Views:

436

Pages:

3
Author:

Bailey Ayala

Question 1 Using the definition of a convergent sequence, prove lim (e^(n+1/n)) = e n→∞ Don't use any theorems about convergent sequences Answer lim an = a convergent if ∀ε>0 n→∞ There exist an, No ∈ N such that |an - a|<ε when n > No Choose ε>0 such that |e^(n+1/n) - e|<ε |e^(n/n) * e^(1/n) - e|<ε |e* e^(1/n) - e|<ε |e|(e^(1/n) - 1)|<ε |e^(1/n) - 1|<ε/e -ε/e < e^(1/n) - 1 < ε/e 1 - ε/e < e^(1/n) < ε/e + 1 ln(1 - ε/e) < 1/n < ln(ε/e + 1) 1/ln(1 - ε/e) > n > 1/ln(ε/e + 1) n>1/ln(ε/e + 1) No> 1/ln(ε/e + 1) So ∀ε>0 we can find and No ∈ N such that |e^(n+1/n) - e|<ε when n>No Hence lim (e^(n+1/n)) = e n→∞ Question 2 Find the eighth term of each sequence. 144, 36, 9, 9/4, ... Answer 1 Write an explicit formula. Look for a simple addition or multiplication pattern between consecutive terms. Here, each term is 1/4 times the preceding term so the term (1/4)^n or 1/4^n is involved: 1/4, 1/4^2, 1/4^3, 1/4^4 To obtain the original sequence, we multiply each term by 576 (since 144 x 4 = 576): 576/4, 576/4^2, 576/4^3, 576/4^4 Hence, the explicit formula is: an = 576/4^n The eighth term is: a8 = 576/4^8 = 576/65536 a8 = 9/1024 An easier way is to multiply 1/4 to each preceding term starting from 9/4: a5 = 9/4 * 1/4 = 9/16 a6 = 9/16 * 1/4 = 9/64 a7 = 9/64 * 1/4 = 9/256 a8 = 9/256 * 1/4 = 9/1024 Result: a8 = 9/1024 Answer 2 Ans: 9/1024 Solution: Use the previous number and divide it by 4 2nd term:144 / 4 = 36 3rd term:36 / 4 = 9 4th term:9 / 4 = 9/4 5th term: 9/4 / 4 = 9/16 6th term: 9/16 / 4 = 9/64 7th term: 9/64 / 4 = 9/256 8th term: 9/256 / 4 = 9/1024 Question 3 Which absolute value inequality is equivalent to the compound inequality 23 ≤ T ≤ 45? F. |T-11| ≤ 34, G. |T-45| ≤ 22, H. |T-24| ≤ 1, I. |T-34| ≤ 11 Answer Remember: |t-Ideal| ≤ Tolerance Ideal = (45+23)/2 = 68/2 = 34 Tolerance = (45-23)/2 = 22/2 = 11 |t-34| ≤ 11 Question 4 How do you find the sum of the geometric sequence 2,4,8...if there are 20 terms? Answer 1 Sum of GS = Sn = (a(r)^n - 1)/r-1 where a is the first term, n the no. of terms and r the common ratio a = 2, n = 20, r = a2/a1 = 4/2 = 8/4 = 2 S20 = 2.(2^20 - 1)/2-1 S20 = 2(2^20-1) = 2097150 Answer 2 Now let us first understand what a geometric series is. A geometric series is a series in which the ratio of any two consecutive terms is the same. Hence a geometric series looks like a + ar + ar² + ar³ + ... + ar^n where a is the first term and r is the common ratio. Now consider the given geometric series 2, 4, 8, ... And the common ratio between two consecutive terms is r = a2/a1 = 4/2 = 8/4 = 2 Hence we have a = 2 and r = 2. Now we know that sum of n terms of GP is given by Sn = a(r^n - 1)/r-1 Hence substituting a = 2 and r = 2 in the above formula we get, Sn = 2(2^n - 1)/2-1 The above equation represents the sum of n terms of the given GP. Now since we want to find the sum of 20 terms we will substitute n = 20. Hence we get, S20 = 2(2^20 - 1)/2-1 => S20 = 2(2^20 - 1)