Thread: Partial Sum Approximation for Alternating Harmonic Series Question

1. 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

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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.

2. Re: Partial Sum Approximation for Alternating Harmonic Series Question

You should be aware of a theorem about alternating series - that when $\displaystyle |a_n|$ is nonincreasing, $\displaystyle \left| \sum_{n=1}^\infty a_n - \sum_{n=1}^N a_n \right| \le |a_{N+1}|$. So you just need to find an N for which $\displaystyle |a_{N+1}|$ is small enough that you can determine $\displaystyle \sum_{n=1}^\infty a_n$ to within 3 decimal places.

- Hollywood