- #1
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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 anything, for all k?
$$
|s_{N+k} - s_N| = | (-1)^{N+2} a_{N+1} + (-1)^{N+3}a_{N+2} + \cdots + (-1)^{N+k+1}a_{N+k} | \lt |a_{N+1}| + |a_{N+2}| + \cdots + |a_{N+k}|
$$
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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