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Let An be equal to a function where n is greater than or equal to 3 and answer the following questions below.
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- Determine the value of f(2) when 2x3 3x5 + 44 46 f(x) 1. f(2) 2. ƒ(2) (Hint: differentiate the power series expan- sion of (x² +42)-¹.) 4 3. ƒ(2) 4. f(2) 5. f(2) = = = = = = X 4² 1 50 1 10 4 25 + 2 25Use the Integral Test to determine whether the seriesE 11 converges or diverges. n=2 n(Inn)s O Diverges; 11 -dx cannot be determined x(Inx) 11 dx = - 00 Diverges; x(Inx) O Diverges; / 11 dx = 00 x(Inx)s 11 -dx 11 Converges; x(Inx) (In2)ª 11 11 Converges; x(Inx) 4(In2)* 00 11 -dx = 11 O Converges; x(Inx) 4(In2)*13.) Find a power series for the function f(x) = = 5 2-6x3*
- Find the power series for f(x) = 8x³ (1-x4)² 2 Hint: First, find the power series for g(x) = 1x41 (Express numbers in exact form. Use symbolic notation and fractions where needed.) 8 in the form S an. n=1 an = Then differentiate.3 (b) Write power series of f(x) = about x = 0. 1- 4.ra) Find the values of x for which the series x' converges. 5n Answer: (b) Find the sum of the series for those values of x . Sum = (your answer should be a function of x)
- The function f(x) = (10x^2)/(1 / 2x)^2 is represented as a power series. Find c1, c2, c3, c4, c5 and find the radius of convergence R for the series.Use comparison test to show that the series E=1 (2) converges. 5n2-п.20 10. Let f(x)= 1+20x (a) Express this function as a power series. (b) Let g(x) = In(1+ 20x). Express this function as a power series using your answer from part (a).