# Estimating the sum of an alternating series by the 40th partial sum

1. Dec 12, 2008

### donald1403

1. The problem statement, all variables and given/known data
Approximate the sum of the alternating series $$\sum_{n=1}^{\infty} \frac{(-1)^{n+1}}{n^4}$$ by the 40th partial sum and estimate the error in this approximation.

I know how to calculute the sum of the alternating series by 4th or 5th partial sum. I don't think this problem wants me to calculate till 40th partial sum.

How can I calculate like 40th partial sum or 100th partial sum?

2. Dec 12, 2008

### Dick

You can bound the error of the 40th partial sum by looking at the absolute value of the 41st term in the series. It's an alternating series.

3. Dec 13, 2008

### donald1403

the absolute value of the 41st term would be 3.54 x 10^(-7) . but it would only mean that the sum of the series will be within 3.54 x 10^(-7). but how can i calculate the sum of the series. this is not for exam or assignment. this is just an example in the textbook. the answer is

S = S40 = 0.9470326439 but they just wrote down the answer but didn't really explain how they get that number? what formula should i use to calculate the sum of the series?

4. Dec 13, 2008

### Dick

Good point. The infinite sum is related to the Riemann zeta function. It's (7/8)*zeta(4). http://mathworld.wolfram.com/RiemannZetaFunction.html That's all tons of fun and stuff, but I don't know any shortcut formula to get the partial sums. I think they maybe did use a computer to compute the partial sum. You can write a short program (preferably) or use a spreadsheet approach if you're desperate.

5. Dec 13, 2008

### donald1403

thanks again, Dick. I just need to make sure they use computer or calculator to calculate the sum. I sure can calculate by hand for few terms but jz wondering if there is any way to calculate. I guess I just have to depend on calculator. anyway, thanks again!