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
$$S_N = \sum_n^N u_n = \sum_n^N \frac{[(-1)^{n+1}]}{n!}$$

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 $r_N = |u_{N+1}|$