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

[donald1403]

Homework Statement

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?

 

[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.

 

[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?

 

[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.

 

[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!