Be it further enacted, That the director of finance, as soon as possible after the total city assessments and the tax rate applicable thereto have been determined, may prepare and send to each taxpayer a bill showing the assessed valuation of all properties subject to taxation, the amount of tax due thereon, the date on which the tax is due and payable, the date on which penalties apply, and information concerning penalties and interest for nonpayment. Such tax bills shall be payable at the office of the department of finance. Failure to send such bills shall in no way affect the validity of a tax, and all taxes shall be due and payable and subject to the same penalties and interest regardless of whether tax bills are sent to taxpayers or not.