In the following exercises, find the Maclaurin series of
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- Observe the function X f(x) = (1+2x)² In order to find the power series for this function, complete the following steps: 1 1-x a. Start with the series Σ. Replace x with (−2x) in this series and k=0 write the corresponding power series for = 1 1+2x b. Take derivative of the series from part (a) above and relate it to the power series for the function 1 (1+2x)²· c. Multiply both sides of the resulting series from above with x, and obtain the series for Write the first four non-zero terms of this series. X (1+2x)² d. What is the radius of convergence for this series? What is the interval of convergence?arrow_forwardFind a power series representation for the function f(x) = 1- 23 O A. (-3)x"+2 O B. En-0 373n+2 O C. (-3)r3n+2 O D. (-3)r3narrow_forwardQ (3): a) Find the power series (Maclaurin or Taylor) to fifth power for the following function: f(x) = e* + e¬*arrow_forward
- Use differentiation to find a power series representation for the function. Make sure the first term in your series is not 0. f(x) (1 + - 5x)2 n = 0arrow_forwardA] Find the Maclaurin infinite series for the function : y(x) = sinh (n² x)arrow_forwardFind a power series for the function Select one: a. O b. (x-1)" m = 0 6"²+1 (x-1)" 6" O d. (x-1)" 6" O c. O e. n=0 6"+1 (x-1)" Ž M=0 n=0 n=0 (-1)" (x-1)" · 6+1 1 7-x centered at 1.arrow_forward
- Consider the function f(x) = Ax tan-(Bx), with A, B # 0. This function can be represented by a Maclaurin series > Cnx". Some, but not all, of the coefficients C, are equal to zero. Answer the following: n=0 O Which power of x is the first one to have a nonzero coefficient? Answer: 8 (11) What is the coefficient for the term you found in (), in terms of A and B? Answer: AB (ii) Which power of x is the second to have a nonzero coefficient? Answer: (iv) What is the coefficient of the term you found in (ii), in terms of A and B? Answer: (V) Which power of x is the third one to have a nonzero coefficient? Answer: (vi) What is the coefficient of the term you found in (v), in terms of A, and B? Answer:arrow_forwardUse the power series f(x)= In (1 – x) = - , for -1sx<1, to find the power series representation for the following function (centered at 0). Give the interval of com k=1 f(7x) = In (1 – 7x) Which of the following is the power series representation for f(7x)? O A. 0 E in (1-(73) O B. (7x) k=1 k=1 C. OD. Σ 00 k=1 k=1 The interval of convergence is (Simplify your answer. Type your answer in interval notation.) 3°C Cloudyarrow_forwardFind a power series for the function Select one: Σ n = 0\ 3m a. Ο Φ. Ο b. Σ(−1)" n = 0 3" 1 Σ n = 0 3" Od. ((-1)* η = 0 4" C. o X + tantilage H = 0 +4" +. tanti) +3" γ r *r* 1 Στατικο Σ +3+1 4" Sx – 15 4x + 11x - 3 centered at 0.arrow_forward
- Let f(x) = 1 + x 1 X Find the power series representation for the function f(x) by completing the following steps: a. First, express the fraction 1¹ as a power series. = X b. Now, express the fraction as a power series. 1-x 1+x x c. The function f(x) 1-x 1-x + 1 is the sum of the two series from parts (a) and (b). Express the function f(x) as a power series. d. What is the interval of convergence and the radius of convergence for this power series?arrow_forwardRefer to imagearrow_forward00 f(x) = Σχ* k=0 = 1 1 and S(x) = n-1 Ext. k=0 The remainder in truncating the power series after n terms is R₁ = f(x) = S(x), which depends on x. a. Show that R₁(x) = x" /(1-x). b. Graph the remainder function on the interval x < 1, for n = 1, 2, and 3. Discuss and interpret the graph. Where on the interval is R, (x)| largest? Smallest? c. For fixed n, minimize |R₁(x)| with respect to x. Does the result agree with the observations in part (b)? d. Let N(x) be the number of terms required to reduce |R₁(x)| to less than 106. Graph the function N(x) on the interval x < 1. Discuss and interpret the graph.arrow_forward
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