Perl 5, 49 bytes

$i=1;$s+=1/++$i*($s+1/$i<1?$_--&&say$i:-1)while$_

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Perl 5, 42 bytes

This prints TSIJMU infinitely:

$i=1;1while$s+=1/++$i*($s+1/$i<1?say$i:-1)

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...or until Try It Online reaches its limit of 128 KiB of output. Exploits that say$i prints that sequence number and then returns 1.

This is seven bytes longer and takes n from stdin:

$i=1;$s+=1/++$i*($s+1/$i<1?$_--&&say$i:-1)while$_

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