p. 662

50.

The series \(TraditionalForm
      \`∑\+\(n = 0\)\%∞\(\((\(-1\))\)\^n\) 1\/\(n!\)\)is alternating so the error of the nth partial sum (Subscript[s, n]) is less than the absolute value of the (n + 1)th term: \(TraditionalForm
      \`\( | s\_n - L | \ \( < \ 1\/\(\((n + 1)\)!\)\)\)\). We want the error to be  < 5×(10^(-6))so we should find n such that (1/((n + 1)!)) < 5×(10^(-6)) or \(TraditionalForm
      \`\(\((n + 1)\)!\)\  > \ 1\/5×10\^6 = 200000\).

Table[(n + 1)!, {n, 1, 10}]

n = 8 will work.

N[(1/((8 + 1)!))]

(Subscript[s, 8])= (∑_(n = 0)^8)(((-1))^n) (1. /(n!))

The true value of the series is (e^(-1)).

L  = N[(E^(-1))]

Abs[(Subscript[s, 8]) -  L]

52.

The alternating harmonic series is \(TraditionalForm
      \`∑\+\(n = 1\)\%∞\((\(-1\))\)\^\(n + 1\)\/n\). The exact sum is ln(2). If we use the approximation \(TraditionalForm
      \`\(s\_n + s\_\(n + 1\)\)\/2 = 
        s\_n + \(1\/2\) \(\((\(-1\))\)\^\(n + 2\)\) a\_\(n + 1\)\)for n = 20 we get:

(Subscript[s, 20])  =  (∑_(n = 1)^20)((((-1))^(n + 1))/n) // N

(Subscript[s, 20]) + (1/2) (1/21)

Log[2] // N

Abs[(Subscript[s, 20]) + (1/2) (1/21)  -  Log[2]]

Using (Subscript[s, 20])for an approximation has an error  < (Subscript[(| a), n + 1]) | :

(1/21) // N

Abs[(Subscript[s, 20]) - Log[2]]