Chapter 7: Q. 1TF (page 594)
Consider the sequence \(\left \{ \frac{1}{k} \right \}_{k=0}^{\infty }\). The associated sequence \(\left \{ S_{n}\right \}_{n=0}^{\infty }\), where
\(S_{n}=1+1+\frac{1}{2!}+...+\frac{1}{n!}\)
, is a sequence of sums. In Chapter \(8\) we will see that this sequence converges to the number \(e\). Evaluate \(S_n\) for \(n = 1, 2, 3, 10\). How close is \(S_10\) to \(e\)?
Short Answer
For the sequence \(S_{n}=1+1+\frac{1}{2!}+...+\frac{1}{n!}\):
\(S_{1}=2\),
\(S_{2}=2.5\),
\(S_{3}=2.667\),
\(S_{10}=2.71858\).
The value of \(S_{10}\) is approximately \(0.0003\) more than the \(e\).