We have the following Limit Comparison Test for improper integrals: Theorem. Suppose f(x), g(x) are two...
(3 points) For each of the following improper integrals, carefully use the comparison test to decide if the integral converges or diverges. All of your 'similar integrands' should be in the form 1/x” for some power p. 2x 1. dx x3 + 1 A similar integrand whose behaviour is known is A. converges B. diverges , so we find that this integral x +4 2. dx x6 - x A similar integrand whose behaviour is known is A. converges B....
R such that f is integrable on every [a,b] (6) Suppose f is a function and a where b> a. Then we define the improper integral eb f(x)dx=lim | b-oo Ja f(x)da, if that limit exists. Assume that f(x) is continuous and monotonically decreasing on [0,00). Prove that Joof exists if and only if Σ f(n) converges. This result is known as the integral test for series convergence.
(1 pt) Test each of the following series for convergence by either the Comparison Test or the Limit Comparison Test. If either test can be applied to the series, enter CONV if it converges or DIV If it diverges. If neither test can be applied to the series, enter NA. (Note: this means that even if you know a given series converges by some other test, but the comparison tests cannot be applied to it, then you must enter NA...
We want to use the comparison test to determine whether (x + 2x 2/5 is convergent. Choose the correct argument. 0 1 ve for all < >1 and 42 The integral is convergent since — (x + 2x)2/5 22/5 0 2/5 /5-1 <0. 2 opent since a 125215 5 to for all z 2 1 and ſº 215 dz = 215–1 <0. his dvorantino e 23275 2 trail 2 2 1 um fio adig do = -0. 1 The integral...
b) first 6 (nl) 33" Question # 6. (6 marks) (a) Use the limit comparison test to determine the convergence of the following: 2n + 5 m32n2 n2+1 3n+n +6 (11) n=1 2n + 5 n3+1 (iv) na + 2n² 3n7 +n +6 nel ni (b) Suppose I have polynomials f(x) and g(x) whose coefficients are all nonnegative. Determine when converges and wherrit dinerges. fin g(n) n=1 2
9. Suppose that we are given the following information about the functions f,g and their deriva- tives and integrals; =4 f(0) = 0 • f(1) = • f'(1) = 2 g(0) = 5 g(1) = 4 • g'(1) =-2 So f(x)dx = 8 5* |(x)dx = 5 Sa f(x)dx = 11 S3 f (x)dx = 6 (d) (5 points) Evaluate Si f(x)dx. (e) (6 points) Evaluate ( f (.5.1 + 4)d.. (f) (6 points) Evaluate, (ثم) (g) (6 points) Evaluate,...
please answer both questions Question # 6. (6 marks) (a) Use the limit comparison test to determine the convergence of the following: X (i) 2n + 5 n2 +1 n° + 2n2 3n4 +n + 6 n=1 n=1 CX 2n + 5 n3 +1 (iv) ni + 2n2 3n7 +n +6 n=1 n=1 (b) Suppose I have polynomials f(x) and g(1) whose coefficients are all nonnegative. Determine when converges and when it diverges. f(n) g(n) n=1
2. (a) Suppose that f is Lebesgue integrable on R. Find the following limit: n sin(x/n f(x) dz. (b) Find the value of the limit in the special case: linn onsin(x/n) n→oo/.oo X(X2 + 1) dx. 2. (a) Suppose that f is Lebesgue integrable on R. Find the following limit: n sin(x/n f(x) dz. (b) Find the value of the limit in the special case: linn onsin(x/n) n→oo/.oo X(X2 + 1) dx.
Please answer it step by step and Question 2. uniformly converge is defined by *f=0* clear handwritten, please, also, beware that for the x you have 2 conditions , such as x>n and 0<=x<=n 1- For all n > 1 define fn: [0, 1] → R as follows: (i if n!x is an integer 10 otherwise Prove that fn + f pointwise where f:[0,1] → R is defined by ſo if x is irrational f(x) = 3 11 if x...
Section 1 — Integration basics and integration techniques 1. Suppose that f(x) and g(x) are continuous functions defined for 0 < x < 4 and that [ f(x) dx = 4 ["f(x) dx = -8 [9(x) dx = 5 ["g(x) dx = -2 Please be extra careful of the bounds in the integrals above. No partial credit will be given. In problems (a-h), either write down the value of the integral, or, write ? if there is not enough information...