use sum of first 3 terms to estimate sum. how good is the estimate? ∞∑ n=2 (-1)^(n-1) * ( 1/ (n-1) )

Answer 1

The sum of the first 3 terms starting with n = 2 is -1/1 + 1/2 - 1/3 = -5/6

For an alternating series, the accuracy is the absolute value of the next term.

error = |1/4| = ± 1/4

Answer 2

t₁ + t₂ + t₃ = -1 + 1/2 - 1/3 = - 5/6. The error bound for an alternating series is the first term excluded from the sum. In this case, the 1^st excluded term is 1/4 so we are guaranteed that -5/6 < S_∞ < - 7/12.

In fact, the infinite series converges to -ln2 ~ -.69, meaning we are within ~ 16% of the actual sum with only 3 terms.