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- Which if the following is a power series representation of the function f(x) = −1+x ? +∞ a. Σ(-x" - x^²+1) b. (x² + x²+¹) +∞ n=0arrow_forwardQ. Find the Taylor's Series for f(x)= ln(x+1) about x=1arrow_forwardBy properties of logarithms, + x In 1- x, = In(1 + x) – In(1 - x). Use this to find a Taylor series for In[(1 + x)/(1 - x)].arrow_forward
- ↑ Use the following information to complete parts a. and b. below. 3 f(x) = -, a = 1 a. Find the first four nonzero terms of the Taylor series for the given function centered at a. OA. The first four terms are −3+3(x-1)-3(x-1)² +3(x-1)³. OB. The first four terms are 3-3(x-1)+3(x-1)²-3(x-1)³. OC. The first four terms are 3-3(x-1) + 6(x-1)²-9(x-1)³. OD. The first four terms are -3+3(x-1)-6(x-1)² +9(x-1)³. b. Write the power series using summation notation. 3(-1)+1 k=0 (x-1) k 00 Oc. Σ 31-1)*(x-1) k=0 00 OA. OCCER 00 OB. 3(-1)+¹(x-1)* k=0 00 OD. Σ 3(-1)k k=0 (x-1)^ į OWD Warrow_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_forwardFA Use the definition to find the Taylor series centered at c = 1 for the function f(x) = Select one: 6 == X O a. O b. O c. O d. O 6 6 X 6 = Σ(-1)" (6x)" n=0 X Σ(-6)” +¹(x − 1)" n=0 6 ÷ = 2 (- X M=0 = 6 Σ(-1)" (x + 1)” n=0 Ž(-6)"x" e. ==6Σ(-1)" (x - 1)" n=0 xlaarrow_forward
- Integrate the given series expansion of f term-by-term from zero to x to obtain the corresponding series expansion for the indefinite integral of f. If f(x) 6x5 1+x6 X f* f(t)dt - n=0 ∞ Σ(-1)" 6x6n+5 n=0arrow_forwardConsider the function in the following image, if 0<x<1, periodic of period 2. Determine the Fourier series of f and sketch the graph of the series.arrow_forwardLet f(x)=e(x-1)^2-1/(x-1)2 for x ≠ 1 and f(1)=1 a. Write the first four nonzero terms and the general term of the Taylor series for e(x-1)^2 about x=1 b. Use the series found in (a) to write the first four nonzero terms and the general term for the Taylor series for f about x=1 c. Determine the interval of convergence for the series given in (b). d. Use the series for f about x=1 to determine if the graph of f has any points of inflection.arrow_forward
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