$$\huge S_1 = 1-1+1-1+1-1 ......$$
$$\huge S_2 = 1-2+3-4+5-6+7.......$$
$$\huge S_3= 1+2+3+4+5+6.......$$

The sum
$$\huge S_1 = 1-1+1-1+1-1 ......$$ = 1/2

|dw:1418222912665:dw|
$$\huge 2S_2= 1/2$$
$$\huge S_2= 1/4$$

|dw:1418223081549:dw|
$$\huge S_3 - S_2 = 4(1+3+4+5+6+7+....)$$
$$\huge 4S_3 = S_3 -1/4$$
$$\huge 3S_3 =-1/4$$
$$\huge S_3 = -1/12$$

Question

$$\huge S_1 = 1-1+1-1+1-1 ......$$
$$\huge S_2 = 1-2+3-4+5-6+7.......$$
$$\huge S_3= 1+2+3+4+5+6.......$$

The sum 
$$\huge S_1 = 1-1+1-1+1-1 ......$$ = 1/2


|dw:1418222912665:dw|
$$\huge 2S_2= 1/2$$
$$\huge S_2= 1/4$$

|dw:1418223081549:dw|
$$\huge S_3 - S_2 = 4(1+3+4+5+6+7+....)$$
$$\huge 4S_3 = S_3 -1/4$$
$$\huge 3S_3 =-1/4$$
$$\huge S_3 = -1/12$$

anonymous · Answer

$$\huge S_1 = 1-1+1-1+1-1 ......$$
$$\huge S_2 = 1-2+3-4+5-6+7.......$$
$$\huge S_3= 1+2+3+4+5+6.......$$

The sum 
$$\huge S_1 = 1-1+1-1+1-1 ......$$ = 1/2


|dw:1418222912665:dw|
$$\huge 2S_2= 1/2$$
$$\huge S_2= 1/4$$

|dw:1418223081549:dw|
$$\huge S_3 - S_2 = 4(1+3+4+5+6+7+....)$$
$$\huge 4S_3 = S_3 -1/4$$
$$\huge 3S_3 =-1/4$$
$$\huge S_3 = -1/12$$