Here's the problem:
"Show that every infinite sequence is the sequence of partial sums of some series."
My work so far:
Let (a[sub:3n0xcm9z]n[/sub:3n0xcm9z]) be the infinite sequence a[sub:3n0xcm9z]1[/sub:3n0xcm9z], a[sub:3n0xcm9z]2[/sub:3n0xcm9z], ..., a[sub:3n0xcm9z]n[/sub:3n0xcm9z], where a[sub:3n0xcm9z]n[/sub:3n0xcm9z] = f(n). Let (B[sub:3n0xcm9z]n[/sub:3n0xcm9z]) be the sequence of partial sums corresponding to the sequence (b[sub:3n0xcm9z]n[/sub:3n0xcm9z]), so (B[sub:3n0xcm9z]n[/sub:3n0xcm9z]) = b[sub:3n0xcm9z]1[/sub:3n0xcm9z], b[sub:3n0xcm9z]1[/sub:3n0xcm9z] + b[sub:3n0xcm9z]2[/sub:3n0xcm9z], ..., b[sub:3n0xcm9z]1[/sub:3n0xcm9z] + b[sub:3n0xcm9z]2[/sub:3n0xcm9z] + ... + b[sub:3n0xcm9z]n[/sub:3n0xcm9z].
My question:
I'm not quite sure how to find a[sub:3n0xcm9z]n[/sub:3n0xcm9z] and b[sub:3n0xcm9z]n[/sub:3n0xcm9z] such that a[sub:3n0xcm9z]1[/sub:3n0xcm9z] = B[sub:3n0xcm9z]1[/sub:3n0xcm9z], a[sub:3n0xcm9z]2[/sub:3n0xcm9z] = B[sub:3n0xcm9z]2[/sub:3n0xcm9z], etc. I tried letting b[sub:3n0xcm9z]n[/sub:3n0xcm9z] = f(n) - f(n + 1). But I ended up with B[sub:3n0xcm9z]1[/sub:3n0xcm9z] = b[sub:3n0xcm9z]1[/sub:3n0xcm9z] + b[sub:3n0xcm9z]2[/sub:3n0xcm9z] = f(1) - f(2) + f(2) - f(3) = f(1) - f(3) which is not equal to a[sub:3n0xcm9z]1[/sub:3n0xcm9z] = f(1). Any hints on candidates for a[sub:3n0xcm9z]n[/sub:3n0xcm9z] and b[sub:3n0xcm9z]n[/sub:3n0xcm9z]?
"Show that every infinite sequence is the sequence of partial sums of some series."
My work so far:
Let (a[sub:3n0xcm9z]n[/sub:3n0xcm9z]) be the infinite sequence a[sub:3n0xcm9z]1[/sub:3n0xcm9z], a[sub:3n0xcm9z]2[/sub:3n0xcm9z], ..., a[sub:3n0xcm9z]n[/sub:3n0xcm9z], where a[sub:3n0xcm9z]n[/sub:3n0xcm9z] = f(n). Let (B[sub:3n0xcm9z]n[/sub:3n0xcm9z]) be the sequence of partial sums corresponding to the sequence (b[sub:3n0xcm9z]n[/sub:3n0xcm9z]), so (B[sub:3n0xcm9z]n[/sub:3n0xcm9z]) = b[sub:3n0xcm9z]1[/sub:3n0xcm9z], b[sub:3n0xcm9z]1[/sub:3n0xcm9z] + b[sub:3n0xcm9z]2[/sub:3n0xcm9z], ..., b[sub:3n0xcm9z]1[/sub:3n0xcm9z] + b[sub:3n0xcm9z]2[/sub:3n0xcm9z] + ... + b[sub:3n0xcm9z]n[/sub:3n0xcm9z].
My question:
I'm not quite sure how to find a[sub:3n0xcm9z]n[/sub:3n0xcm9z] and b[sub:3n0xcm9z]n[/sub:3n0xcm9z] such that a[sub:3n0xcm9z]1[/sub:3n0xcm9z] = B[sub:3n0xcm9z]1[/sub:3n0xcm9z], a[sub:3n0xcm9z]2[/sub:3n0xcm9z] = B[sub:3n0xcm9z]2[/sub:3n0xcm9z], etc. I tried letting b[sub:3n0xcm9z]n[/sub:3n0xcm9z] = f(n) - f(n + 1). But I ended up with B[sub:3n0xcm9z]1[/sub:3n0xcm9z] = b[sub:3n0xcm9z]1[/sub:3n0xcm9z] + b[sub:3n0xcm9z]2[/sub:3n0xcm9z] = f(1) - f(2) + f(2) - f(3) = f(1) - f(3) which is not equal to a[sub:3n0xcm9z]1[/sub:3n0xcm9z] = f(1). Any hints on candidates for a[sub:3n0xcm9z]n[/sub:3n0xcm9z] and b[sub:3n0xcm9z]n[/sub:3n0xcm9z]?