Series (Convergence, determination, and error)

[GuitarStrings]

Homework Statement

Approximate the sum of the series S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n!} by calculating S_10.

Estimate the level of error involved in this problem.

AND

S = \sum(n from 1 to Infinity) \frac{[(-1)^(n+1)]}{n^4}

Approximate the sum of the series by using the 20th partial sum.
Estimate the error involved in this approximation.

Homework Equations

None.

The Attempt at a Solution

Manually found the sum of the series using a GDC.

Error is less than u_{n+1}. So I found u_{n+1}, but that gives the wrong answer for both the cases.

 

[lanedance]

be careful i think you are using n as both the sum variable and the last series term, i think it should be
<br /> S_N = \sum_n^N u_n = \sum_n^N \frac{[(-1)^{n+1}]}{n!}<br />

as its and alternating series with monotonically decreasing term magnitude the error of the sum to n,

If L is the limit of the series, then the error estimate r_N = |L - S_N| should be less than the N+1 term magnitude r_N = |u_{N+1}|