Answer to Question #91364 in Real Analysis for Sajid

Question

Q. Choose the correct answer.
Q. If a_n=∑_(l=1)^n▒〖(□(1/l))〗, then which of the following statement si true about the sequence {a_n} ?
a. Chauchy sequence.
b. Convergent sequence
c. Not a Cauchy sequence
d. Every subsequence of {a_n} is convergent.

Answer 1

c)

This sequence is a partial sum of the harmonic series. It is known that it diverges (see Harmonic series for instance)

and for a natural n:

$$a_{2n}-a_n \geq \frac{1}{2},$$

which means it's not a Cauchy sequence.