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Q1: \text { 4. Let } C([-3,3]) \text { be the vector space of continuous functions } f:[-3,3] \rightarrow \mathbb{R} \text { with the norm of uniform convergence }\|f\|_{\infty}:=\max _{x \in[-3,3]}|f(x)| \text {. } \text { (i) Consider the linear mapping } L: C([-3,3]) \rightarrow \mathbb{R} \text {, } L f:=\int_{-3}^{3} x f(x) d x \text { Prove that the mapping } L: C([-3,3]) \rightarrow \mathbb{R} \text { is continuous. } \text { (ii) Consider the mapping } F: C([-3,3]) \rightarrow \mathbb{R} \text {, } F(f):=\int_{-3}^{3}|x \| f(x)| d x Why you can not use the same strategy as in part (i) to prove that F is[2 Marks]continuous?See Answer
Q2: \lim _{x \rightarrow 0} \frac{\ln \sqrt{x+1}-\tan ^{-1} x}{x} 11. (7 points) Use Taylor series to evaluate the limitSee Answer
Q3:Instructions 1. Create your own example of an alternating series that is (conditionally) convergent, but not absolutely convergent. Your series should be different than any of those in the notes or text examples. 2. Post your infinite series on Discussion Board on Canvas. Give a brief explanation of how you created your series. There are several different approaches you might take. 3. Peer response: look at the post from at least one other classmate and critique the method used to create the series. Do you think the method should work? Alternatively, if you believe your classmate's method works, describe another way the series might have been created.See Answer
Q4:6. 7. 8. · Σ (5(3)" + (3)") n=1 Ans. 13 n=l 8 η n+√n Σ()" n=0 Σ(3) n=0 Need handwritten step wise full solutionSee Answer
Q5:If the tenth term of an arithmetic sequence is 15 and the common difference is 8, find the sum of the first 20 terms. The sum of the first 20 terms of the arithmetic sequence is XSee Answer
Q6:Sequence The sequences below are either arithmetic sequences or geometric sequences. For each sequence, determine whether it is arithmetic or geometric, and write the formula for the term a, of that sequence. (a) 2,-4,8,... (b) -7,-5, -3,... Type O Arithmetic O Geometric O Arithmetic O Geometric term formula 9-0 4-0 808 0+0 0-0 0.0 X 10 $ 12 E GSee Answer
Q7:Find and simplify a formule for a the term of the given sequence. (Hint: Decide what kind of sequence it is, then use information from within the section.) 15. 19, 23, 27, 31. The simplified formula for the term is a- X Ana Reported BORHOSee Answer
Q8:Find the 13th, 0 term of the arithmetic sequence whose common difference is d=-7 and whose first term is a =4. d 8 X 5See Answer
Q9:Find and simplify a formula for a,, the term of the given sequence. (Hint: Decide what kind of sequence it is then use information from within the section.) 4, 20, 100, 500, 2500.... The simplified formula for the term is a,- 8 X D.O G Nokia c ESee Answer
Q10:Find the 58th term of the following arithmetic sequence. 13, 18, 23, 28, 0 X 唱吧 5See Answer
Q11:Series Solutions-Power Series Review: Problem 2 (1 point) Write the sum -3 +5-7+9-11+13-15 +17-19+21-23+25 using sigma notation. The form of your answer will depend on your choice of the lower limit of summation. k=See Answer
Q12:Series Solutions-Power Series Review: Problem 4 (1 point) Find the first four Taylor polynomials about x = 0, and use a graphing utility to graph the given function and the Taylor polynomials on the same screen. In(+6); 20-5. Po(x)= P₁(x) = P2(x) = P3(x) =See Answer
Q13:Series Solutions-Power Series Review: Problem 10 (1 point) Use sigma notation to write the Maclaurin series for the function, e4*. Maclaurin series =See Answer
Q14:LEARNING GOALS: to use the corollary to the Alternating Series Test to find an upper bound on the error in approximating the sum of a convergent alternating series by its Nth partial sum; to use the Alternating Series Test to show a given series converges by verifying the non-alternating part of the general term of the series is positive, decreasing, and has a zero limit. 2. In Question 1(a), we showed n=0 (-1)" 3n+ 1 converged by the Alternating Series Test. Use this fact to find an upper (−1)n+1 1 1 3n+1 bound on the error associated with the approximation n=1 4 7 + 1 10 1 13 Justify your answer.See Answer
Q15:LEARNING GOALS: Understand the difference between absolute convergence, conditional convergence and divergence. 4. Determine whether the following series, absolutely converges, conditionally converges or diverges. You should show whether absolute convergent or not to make a correct conclusion. 00 3n Σ(-1)"; 5n+n n=1See Answer
Q16:1. [-/4 Points] State where the power series is centered. (-1)^(4n-1) n 4ºn! n=1 Need Help? Submit Answer 2. [-/4 Points] n=0 DETAILS LARCALC11 9.8.006. Need Help? Submit Answer Read It State where the power series is centered. (-1)²(x - 577) 9 (9n)! DETAILS LARCALC11 9.8.008. Read It Watch It/n3. [-/4 Points] Find the radius of convergence of the power series. (If you need to use co or -00, enter INFINITY or -INFINITY, respectively.) Σ(-1)7 n+3 n = 0 Need Help? Submit Answer n=0 Submit Answer DETAILS 4. [-/4 Points] DETAILS Find the radius of convergence of the power series. (If you need to use co or -00, enter INFINITY or -INFINITY, respectively.) (-1)" xn 2 Need Help? Read It n=0 Read It (4n)! Nood Help? LARCALC11 9.8.009. Watch It 5. [-/4 Points] DETAILS Find the radius of convergence of the power series. (If you need to use co or -00, enter INFINITY or -INFINITY, respectively.) LARCALC11 9.8.012.MI. Master It LARCALC11 9.8.013. Watch W/n6. [-/4 Points] DETAILS LARCALC11 9.8.018. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is (-1)"+¹(n + 2)x" Need Help? Submit Answer 7. [-/4 Points] DETAILS LARCALC11 9.8.020.MI. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is (5x) ) - (2n)! Need Help? Read It Submit Answer Read It 8. [-/4 Points] DETAILS Need Help? Submit Answer Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the answer is an interval, enter your answer using interval notation. If the answer is a finite set of values, enter you Σ (2n)!()* n=0 Read It Watch It 9. [-/4 Points] DETAILS Master It LARCALC11 9.8.021. Watch It LARCALC11 9.8.024. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) 1st worksheet (20 questions) II do all/n9. [-/4 Points] DETAILS LARCALC11 9.8.024. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval.) (-1)^n!(x-9)" 7" n=0 Ox = 0 07 < x≤9 Ox # 9/7 + Ox + 9 Ox = 9 07 ≤ x < 9 Need Help? Submit Answer n=1 Need Help? Submit Answer 10. [-/4 Points] DETAILS Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is a finite set, enter your answer using set notation.) (-1)"+¹(x-2)^ n6 Read It Need Help? Submit Answer Read It Watch It DETAILS Read It LARCALC11 9.8.025. 11. [-/4 Points] Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is a finite set, enter your answer using set notation.) (x-3)" +1 (n+1)9"+1 Watch It Choose subject LARCALC11 9.8.026. Comment Watch It verify ✔ Cancel/n12. [-/4 Points] DETAILS LARCALC11 9.8.028. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is (-1)+¹(x-9) ng n=1 Need Help? Submit Answer 13. [-/4 Points] DETAILS LARCALC11 9.8.030. Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is Σ (-1)" x8n +7 0-0 8n +7 Need Help? Submit Answer 14. [-/4 Points] n=1 Need Help? Read It Submit Answer Find the interval of convergence of the power series. (Be sure to include a check for convergence at the endpoints of the interval. If the interval of convergence is an interval, enter your answer using interval notation. If the interval of convergence is n!(x + 9)" 1.3.5... (2n-1) Read It Watch It DETAILS n=0 LARCALC11 9.8.038. Read It 15. [-/4 Points] DETAILS LARCALC11 9.8.040. Find the radius of convergence of the power series, where k is a positive integer. (n!)*x^ (kn)!/n16. [-/4 Points] DETAILS LARCALC11 9.8.042. Find the interval of convergence of the power series, where c> 0. (Be sure to include a check for convergence at the endpoints of the interval. Enter your answer using interval notation.) (-1)"+¹(x-c)" nen n=1 Need Help? Submit Answer Σ n=1 17. [-/4 Points] DETAILS LARCALC11 9.8.046. Write an equivalent series with the index of summation beginning at n = 1. (-1)" + 1(n+1)x" Need Help? Read It Submit Answer Read It 18. [-/4 Points] DETAILS LARCALC11 9.8.050. Find the intervals of convergence of f(x), f'(x), f'(x), and [f(x) dx. (Be sure to include a check for convergence at the endpoints of the intervals. Enter your answer using interval notation.) f(x) = Σ (-1)"+¹(x-4)" n=1 n4" (a) f(x) (b) f'(x) (c) F"(x) (d) f(x) dx/n19. [-/4 Points] DETAILS LARCALC11 9.8.052. Find the intervals of convergence of f(x), f'(x), f"(x), and f(x) = (a) f(x) (b) f'(x) (c) F"(x) (d) [1(x) d dx Need Help? Submit Answer (-1)"+¹(x-4)" n Need Help? Read It 20. [-/4 Points] Let g(x) = 1 + 8x + x² + 8x³ + x²+, where the coefficients are C2n = 1 and C2n + 1 = 8 for n ≥ 0. (a) Find the interval of convergence of the series. (Enter your answer using interval notation.) Submit Answer DETAILS (b) Find an explicit formula for g(x). g(x) = d ff(x) dx. ) dx. (Enter your answer using interval notation. Be sure to check for convergence at the endpoints of the interval.) LARCALC11 9.8.078. Read ItSee Answer
Q17:1. [-/4 Points] DETAILS LARCALC119.9.004. Find a geometric power series for the function, centered at 0, by the following methods. 1 f(x) = 5+x (a) by the technique shown in Examples 1 and 2 f(x) = (b) by long division (Give the first three terms.) f(x) = Need Help? Submit Answer л-0 2. [-/4 Points] DETAILS LARCALC11 9.9.006. Find a geometric power series for the function, centered at 0, by the following methods. 6 f(x)= 9-x f(x) = (a) by the technique shown in Examples 1 and 2 Σ (b) by long division (Give the first three terms.) f(x) = Need Help? Submit Answer Read It 3. [-/4 Points] f(x) = Σ Л-0 Read It Watch It DETAILS LARCALC11 9.9.007. Find a power series for the function, centered at c. f(x)= c=2 Determine the interval of convergence. (Enter your answer using interval notation.)/n4. [-/4 Points] DETAILS Find a power series for the function, centered at c. 5 2x C = 4 f(x) = f(x) = n=0 Determine the interval of convergence. (Enter your answer using interval notation.) Need Help? Read It Submit Answer 5. [-/4 Points] f(x) = f(x) = Find a power series for the function, centered at c. 4 3x + 2 n=0 Need Help? Submit Answer g(x) = DETAILS Determine the interval of convergence. (Enter your answer using interval notation.) -Σ LARCALC119.9.012. c=5 6. [-/4 Points] DETAILS n=0 Read It Watch It LARCALC11 9.9.014.MI. Find a power series for the function, centered at c. g(x) = 4x x²+2x-3 Master It LARCALC11 9.9.015. c=0/n7. [-/4 Points] Find a power series for the function, centered at c. 9 f(x) = Σ n=0 Determine the interval of convergence. (Enter your answer using interval notation.) Need Help? Read It Submit Answer 8. [-/4 Points] f(x) = DETAILS LARCALC11 9.9.018. h(x) = Use the power series - h(x) = n=0 DETAILS LARCALC11 9.9.019. to find a power series for the function, centered at 0. -2 1 2²1 Submit Answer c=0 Σ(-1), x < 1 n=0 = Need Help? Read It + Determine the interval of convergence. (Enter your answer using interval notation.) 1 Watch It/n9. [-/4 Points] Use the power series f(x) = n=0 to determine a power series for the function, centered at 0, п-D DETAILS f(x)= Submit Answer 10. [-/4 Points] Need Help? Read It f(x) = 14 (x + 1) Determine the interval of convergence. (Enter your answer using interval notation.) Use the power series n=0 (-1)"x", |x| < 1 DETAILS f(x) = In(x + 1) = LARCALC11 9.9.022. = Σ(-1) ²² x < 1 n=0 to find a power series for the function, centered at 0. Need Help? Read It 7 Watch It LARCALC11 9.9.023. -dx x + 1 Determine the interval of convergence. (Enter your answer using interval notation.) Watch It/n11. [-/4 Points] Use the power series f(x) = (-1)^² x < 1 л-0 to find a power series for the function, centered at 0. f(x) = In(x + 1) Σ Determine the interval of convergence. (Enter your answer using interval notation.) Need Help? Read It Submit Answer DETAILS 12. [-/4 Points] f(x) = Use the power series Σ n=0 LARCALC11 9.9.026. DETAILS LARCALC11 9.9.028.MI. n=0 to find a power series for the function, centered at 0. f(x) = arctan 9x Submit Answer Watch It Σ(-1)^² x < 1 Need Help? Read It Determine the interval of convergence. (Enter your answer using interval notation.) Master It/n13. [-/4 Points] Use the power series f(x) = Σει to find a power series for the function, centered at 0. 1 f(x) = (1-x)² n=1 Submit Answer Determine the interval of convergence. (Enter your answer using interval notation.) Need Help? Read It 14. [-/4 Points] DETAILS Use the fact that (a) (b) - Submit Answer DETAILS LARCALC11 9.9.037. (1-x)" #Σ (7)" -E n=1 ÷ [-(9)*-[ Need Help? Read It Watch It Σ nxn-1 to find the sum of each series. LARCALC11 9.9.042./n15. [-/4 Points] LARCALC11 9.9.047. Find the sum of the convergent series by using a well-known function. (Round your answer to four decimal places.) (−1)″ + 1_ 4"n Need Help? Submit Answer Need Help? Submit Answer 16. [-/4 Points] LARCALC11 9.9.049. Find the sum of the convergent series by using a well-known function. (Round your answer to four decimal places.) 2n (-1)+1; 5"n n=0 DETAILS Need Help? Read It Submit Answer DETAILS Read It 17. [-/4 Points] LARCALC11 9.9.051. Find the sum of the convergent series by using a well-known function. (Round your answer to four decimal places.) 1 (-1) 62n+1(2n +1) Watch It DETAILS Read It Watch It Watch It/n18. [-/4 Points] DETAILS LARCALC11 9.9.052. Find the sum of the convergent series by using a well-known function. (Round your answer to four decimal places.) (−1)″ +1- Need Help? Submit Answer 19. [-/4 Points] f(x) = Σ n=0 Find a power series for the function, centered at c. 5 f(x) = c=-3 5-x Submit Answer 1 g2n-1(2n-1) Determine the interval of convergence. (Enter your answer using interval notation.) Read It DETAILS LARCALC11 9.9.501.XP. Need Help? Read It n-06 20. [-/4 Points] DETAILS LARCALC11 9.9.504.XP. Find the sum of the series. (Round your answer to six decimal places.) (-1)^2+1 2n + Need Help? *(2n + 1)! Read ItSee Answer
Q18:Problem 3. (1 point) Find all the values of x such that the given series would converge. The series is convergent (10x)" n³ from x = left end included (Y,N):. to x = _, right end included(Y,N):. Answer(s) submitted: ⚫ no response ⚫ no response ⚫ no response ⚫ no response submitted: (incorrect) recorded: (incorrect)See Answer
Q19:1. Find explicit formulas for sequences of the form a₁, a2, a3, ... with the initial terms given below: 1/5, 3/20, 5/80, 7/320, 9/1280See Answer
Q20:5. A geometric sequence has a third term of 20 and seventh term of 324/125. Find the two possible common ratios, and the initial term. See Answer

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