Partial Sum Approximation for Alternating Harmonic Series Question
1. The problem statement, all variables and given/known data
Find a value for n for which the nth partial sum is ensured to approximate the sum of the alternating harmonic infinite series to three decimal places.
2. Relevant equations
Sn = Ʃ(-1)^k+1*1/k = 1 - 1/2 + 1/3 - 1/4 + 1/5 - . . .
S1 = 1
S2 = 1 - 1/2
S3 = 1 - 1/2 + 1/3
S4 = 1 - 1/2 + 1/3 -1/4
Sn = ?
3. The attempt at a solution
An attempt was made to derive a formula that would permit finding the value of the nth partial sum (e.g., S1000 = ?), but without success. It appears as though such a formula might not exist. Any help in this regard would be most appreciated.
Re: Partial Sum Approximation for Alternating Harmonic Series Question
You should be aware of a theorem about alternating series - that when is nonincreasing, . So you just need to find an N for which is small enough that you can determine to within 3 decimal places.