Find exactly the value of the integral

Question

 

f(x) =  with an interval of convergence, -3 < x < 3. Find exactly the value of  . Your answer will be a positive integer. Type your answer in the space below (ex. 4)

Answer 1

Since the series for f converges for 0< x <2,

The series can be integrated term by term.

This leads to a general anti derivative term xⁿ⁺¹ / 3ⁿ⁺¹

This series also converges for 0 < x < 2, so

it can be evaluated at the upper ( x = 2) and lower ( x = 0 ) limits.

The lower limit is zero (term by term ) leaving the series

∑ 2ⁿ⁺¹ / 3ⁿ⁺¹

This is a geometric series with common ratio = 2/3

The sum is (2/3) / ( 1 - (2/3)) = 2

Answer 2

∫_{(0 to 2)} f(x)dx = ∫_{(0 to 2)} [(1/3) + (2/9)x + (3/27)x² + ...] dx

= [(1/3)x + (1/9)x² + (1/27)x³ + ...]_{(from 0 to 2)}

= 2/3 + 4/9 + 8/27 + ... This is a geometric series with common ratio r = 2/3. Since r is between -1 and 1, the series converges and has sum a/(1-r), where a is the first term.

So, the sum (i.e., the value of the integral) is 2/3 / (1 - 2/3) = 2