(8) Prove that dt= 1-t n=1 for x e [-a, a],0< a< 1 and deduce from there a power series expansion for -In(1-x) (8) Prove that dt= 1-t n=1 for x e [-a, a],0
For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is convergence and for any n 2 1, - 0<an n+12n(n+1) For natural number n, an = 1+1+3+--+--log n . x dt By use log x = hen x > 0,· w 1 t Prove that the series is convergence and for any n 2 1, - 0
Prove that the series expansion of the exponential function is cauchy 1 e" -Σ n! 321 x 1972-0
12. Use the Maclaurin expansion for e-t to express the function F(2) = dt as an alternating power series in 2. How many terms of the Maclaurin series are needed to approximate the integral for x=1 to within an crror of at most 0.001? Let
-1-1 arctan n n" n!5* (c) Find the interval of convergence and radius of convergence for )0301 i )e-3r) (d) Use the geometric series to write the power series expansion for i. f(1)- 2-4r, centered at a = 0. i.)4 centered at a-6. (e) Write the first 4 nonzero terms of the Maclaurin expansion for i, f(z) = z2 (e4-1) ii. /(x) = cos(3r)-2 sin(2x). (0) Use the Taylor Series definition to write the expansion for f(a)entered at (8) Use...
2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C (b) Prove that when z є R, the definition of exp z given above is consistent with the one given in problem (2a), assignment 16. Definition from Problem (2a): L(x(1/t)dt E(z) = L-1 (z) 2 (1) For z E C, define exp z - n-0 (a) Prove that the infinite series converges absolutely for z E C...
find power series for (1/(1+x^2)). use this power series to prove that the taylor series centered at x=0 for actan(x) is x -x^3/3 +x^5/5 -... (-1)^n ((x^2n+1)/(2n+1))...
8. Use the binomial series with p= -1/2 to get a power series expansion for — valid on (-1,1). Use this to get power series expansions for first VT E, and then arcsin x on this same interval. V1 r2, and then
Use the power series itxË (-1)"X", Ixl < 1 -n=0 to determine a power series for the function, centered at 0, 14 02 7 f(x) (x + 1) dx2 ( x + 1 00 f(x) no Determine the interval of convergence. (Enter your answer using interval notation.) 3. [-17.69 Points] DETAILS LARCALC11 9.2.061. Find all values of x for which the series converges. (Enter your answer using interval notation.) 00 (8x)" n=1 For these values of x, write the sum...
n=0 4. Using the power series cos(x) = { (-1)",2 (-0<x<0), to find a power (2n)! series for the function f(x) = sin(x) sin(3x) and its interval of convergence. 23 Find the power series representation for the function f(2) and its interval (3x - 2) of convergence. 5. +
Use the power series 1 1 + X = Ë (-1)^x), 1x! < 1 n=0 to find a power series for the function, centered at 0. 1 g(x) x + 1 00 g(x) = Σ n=0 Determine the interval of convergence. (Enter your answer using interval notation.)