- #1

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## Homework Statement

Show that [itex]\prod _{n=1} ^{\infty} (n\sin (1/n))[/itex] converges

## Homework Equations

[itex]\prod _{n=1} ^{\infty}a_n[/itex]

**converges**iff the sequence of partial products converges to a non-zero limit. Such a product converges iff [itex]\sum _{n=1} ^{\infty} \log (a_n)[/itex] converges. [itex]\sum _{n=1} ^{\infty} |log(a_n)|[/itex] converges iff [itex] \sum _{n=1} ^{\infty}|a_n-1|[/itex] converges.

## The Attempt at a Solution

Since log(nsin(1/n)) is negative for all n, the product we're interested in converges:

- iff the sequence of partial products converges to a non-zero finite number

- iff the sum [itex]\sum _{n=1} ^{\infty} \log (n \sin (1/n))[/itex] converges

- iff the sum [itex]\sum _{n=1} ^{\infty}| \log (n \sin (1/n))|[/itex] converges

- iff the sum [itex]\sum _{n=1} ^{\infty}|n \sin (1/n) - 1|[/itex] converges

I can't figure out what to compare any of these to to show they converge. Any hints?