a. Write the power series using summation notation.b. Determine the interval of convergence of the series.f(x) = In (1 + 6x)a. Write the power series using summation notation. Choose the correct answer below.00(- 1)k6kO A. 2kk= 1(- 1)k6k+1xk+1O B.k!k= 100(- 1)k+16kО с.k!k= 1(- 1)k+16kO D. 2kk= 1b. The interval of convergence is(Simplify your answer. Type your answer in interval notation.)

Result: The power series expansion of the function ln (1+x) is: ln (1+x)=∑k=1∞(-1)k+1kxk The interval of convergence of the power series is: (-1, 1]The given function is: f(x)=ln(1+6x) Let t=6x. Substitute in 6x=t in the function: ln(1+t)ln(1+t) Write the power series expansion of the function: ln(1+t)=∑k=1∞(-1)k+1ktk where the interval of convergence is (-1, 1]. When t∈(-1,1], ln(1+t)=∑k=1∞(-1)k+1ktkWhen t∈(-1,1], ln(1+t)=∑k=1∞(-1)k+1ktk Substitute t=6x: When 6x∈(-1,1], ln(1+6x)=∑k=1∞(-1)k+1k(6x)k ln(1+6x)=∑k=1∞(-1)k+1k(6x)k Split the powers: ln(1+6x)=∑k=1∞(-1)k+1k6kxk=∑k=1∞(-1)k+16kkxk where 6x∈(-1,1].Result: If a∈(b,c], then b<a≤c 6x∈(-1,1] Hence, -1<6x≤1 Divide the inequality by 6: -16<6x6≤16-16<x≤16 Write the inequality in interval notation: x∈(-16 , 16 ]The power series expansion of the function f(x)=ln(1+6x) is (Option D): ln(1+6x)=∑k=1∞(-1)k+16kkxk where x∈(-16 , 16 ]. The interval of convergence is: (-16 , 16 ]