Mathematics

Problem

The number

e^{-1}

can be written as the alternating series

e^{-1}=\sum_{n=0}^{\infty}(-1)^{n}\,\frac{1}{n!}=1-1+\frac12-\frac16+\frac1{24}-\frac1{120}+\dots.

Let

P_4

denote the fourth–degree truncation (through the

n=4

term) of the series. a. Write

P_4

explicitly and evaluate it to six decimal places. b. Compute the magnitude of the next omitted term,

|a_5|

, and give its decimal value. c. Determine whether

P_4

approximates

e^{-1}

to within

0.0001

. Justify your answer numerically. d. Find the least positive integer

N

such that the magnitude of the

(N+1)

st term is less than

0.00005

. State the value of

N

and explain your reasoning.