Re: Does this series converge or diverge?

In article <0c3761cb-a458-47a5-b3fb-537372c6e881@xxxxxxxxxxxxxxxxxxxxxxxxxxxx>,
nicegirl_130@xxxxxxxxx wrote:
\Sum(n =1, oo) (1 + sin(n))/sqrt(n).

I tried everythiing I know, but nothing worked, total frustration.
Is the fact that sin(n) is dense in [0, 1] useful here?

Since sin(n) is the imaginary part of exp(in), we have from the
formula for the sum of a geometric series

N
---
> sin(n)
---
n=1

N
---
= Im( > exp(in) )
---
n=1

exp(iN) - 1
= Im( exp(i) ----------- )
exp(i) - 1

N+1 exp(iN/2) - exp(-iN/2)
= Im( exp( i --- ) ---------------------- )
2 exp(i/2) - exp(-i/2)

N+1 sin(N/2)
= sin( --- ) --------
2 sin(1/2)

Thus, the sum of sin(n) is absolutely bounded by 1/sin(1/2) < 2.1.
By Dirichlet's Test <http://en.wikipedia.org/wiki/Dirichlet's_test>,
we have that

oo
--- sin(n)
> -------
--- sqrt(n)
n=1

converges. However, by comparison with the harmonic series,

oo
--- 1
> -------
--- sqrt(n)
n=1

diverges. Since the series you are asking about is the sum of a
convergent and a divergent series, it must diverge.

Rob Johnson <rob@xxxxxxxxxxxxxx>
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