- #1

hth

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## Homework Statement

Given the infinite series, [tex]\sum[/tex] n=1 to [tex]\infty[/tex] of sin(nx)/n^(s) is convergent for 0 < s <= 1.

## Homework Equations

## The Attempt at a Solution

Let f_n(x) = sin(nx) and g_n(x) = 1/n^(s)

i.) lim n-> [tex]\infty[/tex] f_n = 0

I'm not sure how to show this formally. Specifically, for a sequence instead of a function.

ii.) [tex]\sum[/tex] |g_(n+1) - g_n| converges.

Or this either.

iii.) [tex]\sum[/tex] g_n, its partial sums are uniformly bounded.

[tex]\sum[/tex] |g_(n+1) - g_n| = (g_1 - g_2) + (g_2 - g_3) + ... + (g_n - g_(n+1) = g_1 - g_(n+1). Lim n->infinity [tex]\sum[/tex] |g_(n+1) - g_n|= lim n->infinity (g_1 - g_(n+1)) = g_1. Therefore the partial sums are bounded, so [tex]\sum[/tex] n=1 to [tex]\infty[/tex] of sin(nx)/n^(s) is convergent for 0 < s <= 1.