
? Wi (2n)! 5^ (n!) Series is given. 9=0 Show if it's Gavergent noto er
1. Given the series -1)" n! , 2n+1 (2n1) (i) Find the radius of convergence of the series. (ii) Find also the largest open interval on which the series converges. 2. (a) Find the Taylor series, in summation form, of f(x) = 1+1 (b) (i) (ii) Find the radius of convergence of the series. Find also the largest open interval on which the series converges. 3. (a) Find two series solutions of the differential equation +9=0, -oo < x <...
Find the interval of convergence of the power series: 5) 00 2n -(4x – 8)" n n=1 E (n + 1)(x - 2)" (2n + 1)! n=0 7) 00 w n(x + 10)" (2n)! n=0
(a) Show that the function defined by the power series 20+1 y=(-1)" 2n +1 n=0 satisfies the differential equation: (1+2?)y = 1. (b) Find the radius of convergence and the interval of convergence of the power series "-3 (x - 3)" 72 nao
(1 point) Use power series to solve the initia-value problem 2n+1 2n Answer: y- n-0
(1 point) Use power series to solve the initia-value problem 2n+1 2n Answer: y- n-0
QUESTION 23 Find the series' radius of convergence. > (x - 2n n=0 0, for all x
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 definition of 0 to show that 5n^5 +4n^4 + 3n^3 + 2n^2 + n 0(n^5).Use the definition of 0 to show that 2n^2 - n+ 3 0(n^2).Let f,g,h : N 1R*. Use the definition of big-Oh to prove that if/(n) 6 0(g{n)) and g(n) 0(h{n)) then/(n) 0(/i(n)). You should use different letters for the constants (i.e. don't use c to denote the constant for each big-Oh).
11. (6 points) Find the sum of the following series: (a) Σ 2n +1 3η n=0 ΟΙ (5) Σ n! ΠΟ
3. The following series is attributed to Newton. It can be used to calculate r. n-0 (n!) (2n 1) 24n+1 (a) (2 points) Prove that the series converges. (b) (2 points) Compare S5 to the actual value of π.
3. The following series is attributed to Newton. It can be used to calculate r. n-0 (n!) (2n 1) 24n+1 (a) (2 points) Prove that the series converges. (b) (2 points) Compare S5 to the actual value of π.
2. (a) Show that the series sin "2n Sman 1 ) converges n = 1 (b) Find an estimate of the magnitude of the error if the sum of the series is calculated by summing up the first 20 terms of the series. [4+3=7 pts]