Prove the Alternating Series Test using Cauchy Criterion.

Question

The Alternating Series Test : Suppose $(a_n)$ is a sequence

1. $a_1 >= a_2 >= a_3 >=...$

2. $lim$ $a_n$=0

Then $\sum_{i=1}^n $$(-1)^n$.$a_n$ converges where n--> inf.

My Attempt:

Given lim $a_n$=0. So for $\epsilon$ > 0 and n >= N, we have |$a_n$| < $\epsilon$.

Consider |$s_n - s_m$| = |$a_{m+1}$ - $a_{m+2}$ + $a_{m+3}$ - $a_{m+4}$ +……$+-$ $a_n$| <= |$a_{m+1}$| + |- $a_{m+2}$| + |$a_{m+3}$| + ….. + |+-$a_n$| < (n - m).$\epsilon$, whenever m, n >= N.

Put $\epsilon_0$ = (m + n).$\epsilon$ and we get |$s_n - s_m$| < $\epsilon_0$

Is this correct?

Thanks in advance :)

Answer 1

For m,n > l, |$s_n - s_m$| = |$a_{m+1} - a_{m+2} + a_{m+3} - … +-a_n$| <= $a_l$ .....(1)

Since $lim a_n$ = 0, there exists N s.t |$a_l$| < $\epsilon$ whenever l >= N.....(2)

From (1) and (2),

|$s_n - s_m$| < $\epsilon$