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.
39. p(x) = 2x6 ln (1 − x)
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Calculus: Early Transcendentals (2nd Edition)
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- Find a power series for the function, centered at c. 4 f(x) = 3x + 2' f(x) = Determine the interval of convergence. (Enter your answer using interval notation.) (#) 5 13arrow_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_forwardQ// Consider the two series such that: f(x) = 1 + 2x + 3x2 +4x3 + ... and g(x) = 1 + 2x + 3x2 +4x3 + a. Find the sum of the two generating functions. Then find the generating function for the result. b. Find the product of the two generating functions. Attach File Browse My Computerarrow_forward
- Use the power series to find a power series for the function, centered at 0. -2 x² 1 h(x) = ∞ 1 ₁ + x = (-1)"x", |×| < 1 1 n = 0 h(x) n = 0 = = 1 1 + x + 1 1 - X Determine the interval of convergence. (Enter your answer using interval notation.) |(−1,1)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_forwardwrite expansion of function as a power seriesarrow_forward
- sum Sn. (b) Use (a) to determine whether the series is convergent or divergent. 8 (5) Decide whether each of the following statements is true or false. If a statement is true, explain why. If a statement is false, provide specific examples of ak and Σb for k=1 which the statement is false. Σ (a) If 5 am is a series such that ak a k=1 1 k² < ak for all k, then (d) If ak k=1 (b) If ak is a series such that 0 < aarrow_forwardhow do i solve the attached calculus question?arrow_forwardH.W 1/ For the following series: - 1 (x-2), (x-2)2 (x-2)" 21 S(x) +(-1)". Find: - a- Values of (x) that make series converged b- The sum of series Ans/ a) 0arrow_forwardEvaluate the infinite series by identifying it as the value of an integral of a geometric series. (- 1)" 57+ (n + 1) 8. n+1 1 | f(t)dt where f(x) = (– 1)" Hint: Write it as 5 + I n=0arrow_forward00 1 Use the equation 2x" for |x| < 1 to expand the function in a power series with center c = 0. 1 - x 1- x4 n=0 (Express numbers in exact form. Use symbolic notation and fractions where needed.) Σ 1- x n=0 Determine the interval of convergence. (Give your answer as an interval in the form (*,*). Use the symbol co for infinity, U for combining intervals, and an appropriate type of parenthesis "(",")", "[","]" depending on whether the interval is open or closed. Enter Ø if the interval is empty. Express numbers in exact form. Usc symbolic notation and fractions where needed.)arrow_forwardshow full and complete procedurearrow_forwardarrow_back_iosSEE MORE QUESTIONSarrow_forward_iosRecommended textbooks for you
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