Combining power series Use the power series representation
to find the power series for the following functions (centered at 0). Give the interval of convergence of the new series.
35. f(3x) = ln (1 − 3x)
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Calculus, Single Variable: Early Transcendentals (3rd Edition)
- Use the power series = (-1)"x" to determine a power series centered at 0 for the function f(x) = - %3D (8x + 1)2 8r +1 O Eo(-8)"(n– 1)x"-1 O Eo(-8)"zx"-1 O Eo(8)">x" O Eo(8)"zx"-1 O Eo(-8)"x"arrow_forwardQ/ for @ ER, use the power series for the exponential function e² to show that ele (-1)⁰ (2n)! 2n=1 + Po (-1) 82u+1 (2n+1)!arrow_forwardx²- Q6/ Find five terms of Maclaurin series for f(x) = (1 +x)K where (k) is any Real number, then find exact and approximate value of function if x = 1, k= 2 Ans./ f(x) = 1+ kx + k(k-1) k(k-1)(k-2) „3 1 k(k-1)(k-2)(k-3) ,4 2! 3! 4! @ x = 1 & k = 2 - Exact = 4 Аpproximate 3 4 Q7/ a) Find the first four nonzero terms of Maclaurin series for f (x) = e-x² b) Find the approximate value of the function in part (a) at x = and compare it with exact value? 2x2 12x 120x6 Ans./ a) f(x) =1- 2! +... 4! 6! 2 - Exact = 0. 64118 3 b) @ x == Approximate = 0. 6396 Q8) Considered that the function f(x) = x.e* i. Find the first three nonzero terms in Maclaurin series Compute the approximate value of function (Maclaurin series) at x = 1 and compare it with the exact value? Compute the approximate integral value of the function (Maclaurin series), ii. iii. Ans./ i)f(x) = x + x² +;x³ + ... ii) @ x = 1 - Approximate = 1+ 12 +1³ = 2.5 Exact = 1* e' = 2.7 iii) Integral = 0.958 Q9/ Answer the following points i.…arrow_forward
- 00 For |r| n(ar)" = ar + 2(ar)² + 3(ar)³ + ... = ar Use this to find the power series for f(x) = •.. - -x (1 – ar)? (1+x) n=1 Select the correct answer below: O 2 n(x)" n=1 00 O 2 n(-x)2" n=1 (-x)" n п n=1 O > n(-x)" n=1 (x)" п n=1 00 O 2(-nx)" n=1arrow_forward49arrow_forward00xk Use the power series f(x) = In (1-x) = - Σ, for -1arrow_forward2+1 Recall that sin(x) = Σ(-1)" (2n+1)! Obtain the term of the power series of sin(x + 1) where n = 3 for a value of x = 3.20.arrow_forwardUse the power series 1 1 + x f(x) = n = 0 to find a power series for the function, centered at 0. f(x) = In(x + 1) 80 ∞ n = 0 (−1)"x", |x| < 1 Determine the interval of convergence. (Enter your answer using interval notation.) (-1,1)arrow_forwardLion Use the power series f(x) = Select one: O a. O 4 (4x + 1)² C. 8 O b. Σ (4)" ·n.x² − 1 n=0 1+x d dx - n=0 Σ(-4)" ·n-x²-1 d. Ž (4)".x" n = 0 Σ(-4)" · (n − 1) x²-1 n=0 Time left 1:2 Σ(-1)"x" to determine a power series centered at 0 for the function x ○ e. Ž (-4)”.x² n=0 n=0 4x+1arrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_ios
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