Show directly that 1 11 1n is a Cauchy sequenceSolutionFor

Show directly that (1 + 1/1! +... 1/n!) is a Cauchy sequence.

For all integers n > m ,
|s(n) - s(m)|
= 1/n! + 1/(n-1)! + ... + 1/(m+1)!
< 1/2^n + 1/2^(n-1) + ... + 1/2^(m+1), assuming that m > 3
< 1/2^(m+1) + 1/2^(m+2) + ...
= (1/2^(m+2)) * 1/(1 - 1/2), infinite geometric series
= 1/2^(m+1).

So given > 0, choose a positive integer N > 3 such that 1/2^(N+1) < .
Then, for all m > n > N:
|s(n) - s(m)| < 1/2^(m+1) < 1/2^(N+1) < , as required.