8. If b; = 1+ sin(k/2) then the sequence {b;} has a convergent subsequence. k+1 9. If Ck arctan k + k · sin() then the sequence {Ch} has a convergent subsequence.

Q: Use series to evaluate the limits

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Q: (sin z)2; 20 = 0 20

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Q: Using the nth term test for divergence, the series 2n n +5 an n31 n-1 is divergent since lim an =

A: According to nth term test for divergent, The given series is,

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A: Here we have to use Ratio Test.

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A: Consider the series ∑n=1∞xlnn We have to find positive value of x for which the given series is…

Q: Find the first four nonzero terms of the power series for the function. arctan(2.x) 9. g(x) =

A: #Question 9

Q: 14) (9.) Use the definition of Taylor series to find the series for f(x) = cos(3x) with center 0.…

A: Given : f(x) = cos(3x)

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A: Convergence of Improper Integrals at left-end point: Let us suppose that x=a be the point of…

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Q: Determine if the following converges or dive rges. series ncln nje N3D2

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Q: 1- Find &, and Ea of sin (x) for x = 2 with 1-8 terms of the series. sin(x) - Σ k-0 (-1)* (2k + 1)!…

A: Given : sinx=∑k=0∞-1k2k+1!x2k+1  =x-x33!+x55!-. . . . . . To Find : et   ,  ea Note : According to…

Q: (-1)"n" sin" (4) 2n n=1 CS Samed with Camacarner

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Q: -1+ cosx - Use the Maclaurin Series for cosx to find the following limit: lim

A: To determine the limit using the Maclaurin Series for cos x.

Q: The nth term of the series sin- n=1 has zero limit O has a limit of (+1) O has no limit O has a…

A: Note : The sin(angle) will be always positive if the angle lies in first and second quadrant.  The…

Q: The value of x for which the series - n³(x-3)" diverges will be given by

A:

Q: b, Find the sum of the series (x + 1)" n=2 and provide one of the intervals on which it converges.…

A: You have asked multiple unrelated questions in the same post. I have addressed the first one. Please…

Q: Find Taylor series at x = 0 for the functions sin πx

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Q: 10. Suppose that the function by the power series 3 X f(x)=1= =+ = + ... + (−1) 2 4 8 Find the…

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Q: For the function f(x) find the power series representation and its interval of (1 – 4.x)²…

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Q: V3 is V3 4. The series E=1(sin- - sin¬ -1 n+1. n+2, A) Convergent to – 3 B) Convergent to 2 C)…

A: Answer: None of them.

Q: 3. For what values of x , does the series, + converge ?Find the ........... first two derivatives of…

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Q: The Taylor series about the point c = 1 for the function f(x) = arctan(x) 4 is given by Σα(-1)". n=1…

A: We have to find the value of a2 from the Taylor series.

Q: Obtain the 5th term of the power series of sin(x) for a value of x = 4.55.

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Q: 3) *It can be shown that the improper integral e* cos xdx : Can we conclude that 2 -- cos n…

A: use integral test for checking convergence.

Q: se series to evaluate the limit. 1 – cos(4x) lim x →0 1 + 4x – e4×

A:

Q: How many terms are needed in series of cos x to compute for |x| < (1/2) accurate to 1/2 decimal…

A: The (minimum) number of terms needed to compute the values with maximum error =0.1,0.01

Q: f (x) = arctan(r²)

A: We know that power series of arctan(x)

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Q: 2. (a) Use Equation 1 to find a power series representation for f(x) = In(1 – x). What is the radius…

A: Log (1-x) =-integration of 1/1-x According to guidelines we solve only first question.

Q: The radius of convergent for the series (a)co (b))

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Q: Find the first three nonzero terms of the Maclaurin series for the function and the values of x for…

A: Solution

Q: 1. Find the interval and radius of convergence 00 (4x – 2)" (-1)" (4n + 1) n=0

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Q: 8. Use the fact that ()1 converges to 0 along with the theory derived in this section and worksheet…

A: We can solve this by the help of squeeze theorem and given details

Q: . Consider the function f(x) = cos x − 3x + 1. Since ƒ (0)ƒ (²) < 0, ƒ (x) has a root in [0]. To…

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Q: Find Taylor series at x = 0 for the functions cos (x5/3)

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Q: The series ∑Cn(x+9)^n converges at x=−4 and diverges at x=−17.

A: We know that if the power series ∑ n=1∞an (x-b)n is convergent at x=a, then this series is…

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Q: (1) Use the Maclaurin series of sin x to evaluate the limit ,3 sin x – x +° lim

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Q: Give McLaurin Series of Cimmon function for sinh x

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Q: 00 Σ 6. a, where a, = sin" Consider the series n=1 Compute the limit: lim yan = (Enter "infinity" or…

A: ∑n=1∞ an here we have an=sinn6n3 use the square root test L=limn→∞an1n=limn→∞ann if L&lt;1 then the…

Q: 3. If the series Eo akx* converges to the function f(x) on the interval (-2, 2), provide a formula…

A: Given:  So, the nth derivative is:

Q: Use sigma notation to write the Taylor series about x = 8 for the function. 1 x + 5 k = OW

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Q: What is the function represented by the power series from k=0 to infinity of 2^k*x^(2k+1)?

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Q: Using the nth term test for divergence, the series a, = 32n+1 n=1 7n=1 is divergent since lim an=

A: "Since you have asked multiple question, we will solve the first question for you. If you want any…

Q: Use series to evaluate the limits

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Q: 1) Let f(x) = In(1 + x) . Show that the Maclaurin series for f is E,(-1)"+1 . Hint: take 4…

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Q: \n + 2, The following telescoping series ΣΗ 1 sin sin-1 n + 1 n + 2, n=1 TT converges to converges…

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Q: Find the sum of the convergent series.

A: Given, ∑n=1∞sin1n S=sin1+(sin1)2+sin13+.... it is a geometric series where: a=sin1,  r=sin1 we know…

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Q: Using the nth term test for divergence, the series Σ 32n+1 n=1 n=1 is divergent since lim an

A: