- #1
TeenieBopper
- 29
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Homework Statement
Prove [itex]\sum\frac{(-1)^k}{k^2}[/itex] is a Cauchy sequence.
Homework Equations
Definition of Cauchy sequence: |a[itex]_{n}[/itex] - a[itex]_{m}[/itex]|<ε for all n,m>=N, n>m
The Attempt at a Solution
I thought if I could prove that the above summation was less than the summation of 1/k^2, the problem would be easy. But either that isn't true, or I don't know how to prove it.
So I started some stuff with the triangle inequality. Here's what I have in full so far:
Let ε>0 be given. Choose a positive integer N such that 1/N<ε. Then, if m,n>=N, and n>m, we have |a_n - a_m| <= |[itex]\sum(-1)^n/n^2[/itex]| + |-[itex]\sum(-1)^m/m^2[/itex].
But I don't really know where to go from here. I could say that |a_m| = |-a_m|, but I don't think that gets me anywhere. Any hints to get me going again would be greatly appreciated.