A Technical Lemma

and

Then the sum

where

In particular,

**Proof.** First, observe that for any integers 0 < *x* < *y*, the sum

where

(This can be seen by comparing the sum with the integral .)

Next, since
,
note that

(23) |

Thus, using (21),

(24) |

where

(25) |

Combining this with (22), the error can be bounded by

(26) |

Setting ,

(27) |

Finally, note that since , the sequence . Thus , and