Calculus Questions Related to Power Series

Question

Rewrite the power series  so that its lower index of summation is n = 2. 

 

Which of the following functions equals the power series  ? 

f(x) =  with an interval of convergence, -4 < x < 4. Find exactly the value of  . Your answer will be a positive integer. Type your answer in the space below (ex. 4). 

Answer 1

A. ∑_n=2^∞ (n-1) x^n-2

B.= e^x

C. = f (x) =( 1/4 ) + (2x / 4² ) + ( 3x² / 4³ ) + ( 4x³/ 4⁴ ) + ( 5x⁴/ 4⁵) +....

Then f (x) =( 1/4 ) + (2x / 4² ) + ( 3x² / 4³ ) + ( 4x³/ 4⁴ ) + ( 5x⁴/ 4⁵) +....

Hence ∫₀² f(x) d x = ∫₀² ( 1/4 ) d x + ∫₀² (2x / 4² ) d x + ∫₀² ( 3x² / 4³ ) d x +....=

= 1/2 + 1/4 + 1/8 + 1/16 +.... = (1/2) /[ 1 -(1/2) ] = 1