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1. Find 7(5") 3" 5n+12n+1 sin(n!) lim (a) im (b) n 2018 пH0 п2 n oo...
3. Find: 7T (1) lim n sin (3) lim arcsin G n-00 n 100 COS- n...
3. Find: 7T (1) lim n sin (3) lim arcsin G n-00 n 100 COS- n n00 n 1 (4) lim (1+ (6) lim (n+1 n) n- 3n n-00 1/n (2) lim arctann Vn2 - 1 (5) lim 2n In (1+) (8) lim n (11) lim n+ n 2+1 (14) lim n- 75n+2 (9) lim (nt n-00 700 n->00 n (7) lim V (Vn+1- Vn) (-1)"n (10) lim n+ on+1 (13) lim (3" +5")1/n sinn (12) lim arctan 2n 2n...
Q11 (Variant of Wallis product). For every integer n 2 0, we define Im r sin dx (a) Show that In+...
Q11 (Variant of Wallis product). For every integer n 2 0, we define Im r sin dx (a) Show that In+,-n+21n (b) Show that 0< 12n+2 S I2n+i < 12n- (c) Use (a) and (b), show that lim Pni1. (d) Repeatedly using (a), show that I2nl2 (e) Compute the limit lim (Historical remark: Q9 and Wallis Product was one of earliest approaches to 22n(n)! 24n(n!)4 (2n+1)!(2n)! 2-n(n!)2- n→oo v 2n+1(2n)!' 22n (n!)2 and then lim approximate π from rationals. This...
1. Find lim f(x) and lim f(x) for each of the following. Assign oo or -...
1. Find lim f(x) and lim f(x) for each of the following. Assign oo or - where appropriate: (a) f(z)=42 -2 (b) f(x)= 3r-2 (e) f(1)--5rt6 (x - 2)2 2. Find each of the f limits, assigning oo or-o where appropriate x+2z-1 (b)lm 3r-2 3 r-2 (c) li 42+1 (k) im (2r V4r2-8r 3) 15r-2 3. Find the horizontal and vertical asymptotes of each of the following functions 4. Sketch the graph of each of the following functions and determine,...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all inte...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
2.13.2 Compute lim sups and lim infs for the following sequences (a) {(-1)"n} (b) {sin (nT/8)}...
2.13.2 Compute lim sups and lim infs for the following sequences (a) {(-1)"n} (b) {sin (nT/8)} (c) {nsin (nt/8)} (d) {[(n + 1) sin (n7/8)]/n} (e) {1+(-1)"} (f) {rn} consists of all rational numbers in the interval (0,1) arranged in some order.
Which of the following series converge conditionally? n nt Σ(-1), sin n -1) n2 tan -1)"ne (2n-1 3-5-7-(2n 1) 3-5 1.3-1.3.52 + (-1) 3 3.5 3.5.7 3+3.5-3-5.7 + Which of the following series con...
Which of the following series converge conditionally? n nt Σ(-1), sin n -1) n2 tan -1)"ne (2n-1 3-5-7-(2n 1) 3-5 1.3-1.3.52 + (-1) 3 3.5 3.5.7 3+3.5-3-5.7 +
Which of the following series converge conditionally? n nt Σ(-1), sin n -1) n2 tan -1)"ne (2n-1 3-5-7-(2n 1) 3-5 1.3-1.3.52 + (-1) 3 3.5 3.5.7 3+3.5-3-5.7 +
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for...
#s 2, 3, 6
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
Question 1 1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2....
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
number 4 1. Find the limit of the following sequences (find lim an) n n a.)...
number 4
1. Find the limit of the following sequences (find lim an) n n a.) an = n +3 b.) an = V35n n- 2. Determine whether the following series converge or diverge. -3 (n + 2)n + 5 b.) tan-'(n) n2 + 1 a.) 5 nel 3. Determine the radius of convergence and the interval of convergence of the series 2" (x – 3)" n n=1 n=0 (-1)", 2n 4. Using the power series cos(x) (2n)! (-« <...
= 7. Determine whether the sequence an find the limit. (2n)3 +sin(n) n+n2 +6 converges or...
= 7. Determine whether the sequence an find the limit. (2n)3 +sin(n) n+n2 +6 converges or diverges. If it converges,
3. Find: 7T (1) lim n sin (3) lim arcsin G n-00 n 100 COS- n n00 n 1 (4) lim (1+ (6) lim (n+1 n) n- 3n n-00 1/n (2) lim arctann Vn2 - 1 (5) lim 2n In (1+) (8) lim n (11) lim n+ n 2+1 (14) lim n- 75n+2 (9) lim (nt n-00 700 n->00 n (7) lim V (Vn+1- Vn) (-1)"n (10) lim n+ on+1 (13) lim (3" +5")1/n sinn (12) lim arctan 2n 2n...
Q11 (Variant of Wallis product). For every integer n 2 0, we define Im r sin dx (a) Show that In+,-n+21n (b) Show that 0< 12n+2 S I2n+i < 12n- (c) Use (a) and (b), show that lim Pni1. (d) Repeatedly using (a), show that I2nl2 (e) Compute the limit lim (Historical remark: Q9 and Wallis Product was one of earliest approaches to 22n(n)! 24n(n!)4 (2n+1)!(2n)! 2-n(n!)2- n→oo v 2n+1(2n)!' 22n (n!)2 and then lim approximate π from rationals. This...
1. Find lim f(x) and lim f(x) for each of the following. Assign oo or - where appropriate: (a) f(z)=42 -2 (b) f(x)= 3r-2 (e) f(1)--5rt6 (x - 2)2 2. Find each of the f limits, assigning oo or-o where appropriate x+2z-1 (b)lm 3r-2 3 r-2 (c) li 42+1 (k) im (2r V4r2-8r 3) 15r-2 3. Find the horizontal and vertical asymptotes of each of the following functions 4. Sketch the graph of each of the following functions and determine,...
In questions 1-8, find the limit of the sequence. sin n cos n 2. 37 /n sin n 3. 4. cos rn 5. /n sin n o cos n n! 9. If c is a positive real number and lan) is a sequence such that for all integer n > 0, prove that limn →00 (an)/n-0. 10. If a > 0, prove that limn+ (sin n)/n 0 Theorem 6.9 Suppose that the sequence lan) is monotonic. Then ta, only if...
2.13.2 Compute lim sups and lim infs for the following sequences (a) {(-1)"n} (b) {sin (nT/8)} (c) {nsin (nt/8)} (d) {[(n + 1) sin (n7/8)]/n} (e) {1+(-1)"} (f) {rn} consists of all rational numbers in the interval (0,1) arranged in some order.
Which of the following series converge conditionally? n nt Σ(-1), sin n -1) n2 tan -1)"ne (2n-1 3-5-7-(2n 1) 3-5 1.3-1.3.52 + (-1) 3 3.5 3.5.7 3+3.5-3-5.7 +
Which of the following series converge conditionally? n nt Σ(-1), sin n -1) n2 tan -1)"ne (2n-1 3-5-7-(2n 1) 3-5 1.3-1.3.52 + (-1) 3 3.5 3.5.7 3+3.5-3-5.7 +
#s 2, 3, 6
2. Let (En)acy be a sequence in R (a) Show that xn → oo if and only if-An →-oo. (b) If xn > 0 for all n in N, show that linnAn = 0 if and only if lim-= oo. 3. Let ()nEN be a sequence in R. (a) If x <0 for all n in N, show that - -oo if and only if xl 0o. (b) Show, by example, that if kal → oo,...
Question 1
1. [5 pts] Give a complete definition of lim f(x) = -oo if... 2. [25 pts] Give an example of each of the following, or state one or more theorems which show that such an example is impossible: a. A countable collection of nonempty closed proper subsets of R whose union is open. b. A nonempty bounded subset of R with no cluster points. c. A convergent sequence with two convergent subsequences with distinct limits. d. A function...
number 4
1. Find the limit of the following sequences (find lim an) n n a.) an = n +3 b.) an = V35n n- 2. Determine whether the following series converge or diverge. -3 (n + 2)n + 5 b.) tan-'(n) n2 + 1 a.) 5 nel 3. Determine the radius of convergence and the interval of convergence of the series 2" (x – 3)" n n=1 n=0 (-1)", 2n 4. Using the power series cos(x) (2n)! (-« <...
= 7. Determine whether the sequence an find the limit. (2n)3 +sin(n) n+n2 +6 converges or diverges. If it converges,
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