Question

ntered at x = 0.) f(x) = x 1 − x5Find a power series representation for the function; find the interval of convergence. (Give your power series representation centered at x = 0.) f(x) = x 1 − x5

Answer 1

Answer:

1/5.

Explanation:

Note that the sum of infinite geometric series is equally to: a/ (1 - r). Where a = first term and r is the common ratio.

Hence, if f(x) = 3/(1 - 5x) ^2. Therefore, taking the another derivative of f(x)` = 30/ (1 - 5x)^3.

Also, f(x)`` = 450/ ( 1 - 5 )^4 and f(x)``` = 9000( 1 - 5x)^5 and f(x)```` = 225000/( 1 - 5x)^6.

If we are to use the Taylor's series formula then;

F(0) + f'(0) + f"(0)/2! × x^2 + f'''(0)/3! × x^3.

[ NOTE: f(0) = 3, f''(0) = 30, f'''(0) = 9000, f''''(0) = 225000].

450/2! = 225, 9000/ 3! = 1500 and 225000/4! = 9375..

Which gives; 3 + 30x + 225x^2 + 1500x^3 + 9375x^4. The ratio test will then give us our answer as 1/5.

As stated above, alternatively we can make use of the sum of geometric series where |x| < 1/5