Resistors in dc circuits:

Components are said to be in series when they are connected end-to-end providing only one path for the current.  Thus the same current passes through all the components (including the power supply).  See diagram below.

When a current flows through a resistor (or a component having resistance) there is a potential difference between its ends.  Thus where two or more resistors are connected in series the potential difference between the extreme ends is the sum of the individual potential differences.

Hence E = V1 + V2 + V3

But from Ohm's Law V = IR

Therefore E  =  IR_TOTAL

So     V1 = IR1             V2 = IR2             V3 = IR3

Thus IR_TOTAL      =  IR1 + IR2 + IR3

                            =  I (R1 + R2 + R3)

So R_TOTAL           =  R1 + R2 + R3