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- Homework Statement
- Given a sequence ##(a_n)## with properties:

1) ##(a_n)## is a decreasing sequence, and ##a_n \gt 0## for all ##n##

2) ##\lim (a_n) = 0##

then the alternating series ##\sum_{n=1}^{\infty} (-1)^{n+1} a_n## converges.

- Relevant Equations
- To prove the above assertion we're asked to show that the sequence of partial sums

## s_n = a_1 - a_2 + a_3 - a_4 + \cdots (-1)^{n+1} a_n##

is a Cauchy sequence.

For any fixed ##N##, we have

$$

|s_{N+k} - s_N| = | (-1)^{N+2} a_{N+1} + (-1)^{N+3}a_{N+2} + \cdots + (-1)^{N+k+1}a_{N+k} | \lt |a_{N+1}| + |a_{N+2}| + \cdots + |a_{N+k}|

$$

Though, from ##\lim (a_n) = 0## we can establish ## n \geq N \implies |a_n| \lt \varepsilon##, that is all the individual terms in the RHS of the above inequality is less than ##\varepsilon##, yet by increasing ##k## the number of ##\epsilon##'s will increase and hence we won't be able to bound it.

How can we show ##|s_{N+k} - s_N|## to be less than

$$

|s_{N+k} - s_N| = | (-1)^{N+2} a_{N+1} + (-1)^{N+3}a_{N+2} + \cdots + (-1)^{N+k+1}a_{N+k} | \lt |a_{N+1}| + |a_{N+2}| + \cdots + |a_{N+k}|

$$

Though, from ##\lim (a_n) = 0## we can establish ## n \geq N \implies |a_n| \lt \varepsilon##, that is all the individual terms in the RHS of the above inequality is less than ##\varepsilon##, yet by increasing ##k## the number of ##\epsilon##'s will increase and hence we won't be able to bound it.

How can we show ##|s_{N+k} - s_N|## to be less than

*anything, for all k?*
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