Given that the series the summation from n equals 1 to infinity of the quotient of negative 1 raised to the n plus 1 power and n is convergent, find a value of n for which the nth partial...

Question

Given that the series  is convergent, find a value of n for which the nth partial sum is guaranteed to approximate the sum of the series with an error of less than 0.0001

A	9
B	99
C	999
D	9999

Answer 1

The obvious answer is D, as the alternating nature of the sum guarantees that the error is less than the next term.

A more sophisticated analysis would show that 5000 is sufficient. That is because we can rewrite the remainder, up to a sign as 1/(n+1)(n+2) + 1/(n+3)(n+4) + ...which is less than 1/2 the integral of 1/k^2 from k = n to infinity = 1/2n