Help me on this problem

ln2=1-1/2 + 1/3 ...
if we rearrange terms we get

(1 + 1/3 + 1/5 + 1/7 ......) + (-1/2 -1/4 -1/6 ........)

separating terms we get
(1 + 1/3 + 1/5 + 1/7 ......) -1/2(1 + /2 + 11/3 + 1/4 ..............)
(1 + 1/3 + 1/5 + 1/7 ......) -1/2(1 + 1/3 +1/5 +1/7 ......+ 1/2 + /4 + 1/6.......)

(1 + 1/3 + 1/5 + 1/7 ......) -1/2(1 + 1/3 + 1/5 + 1/7 ......) -1/2(1/2 + /4 + 1/6.......)

1/2(1 + 1/3 + 1/5 + 1/7 ......) -1/2(1/2 + /4 + 1/6.......)
taking 1/2 common
1/2 (1 -1/2 +1/3 -1/4 .....)

1/2(ln2)




we get ln2 = 1/2 ln2

Such things could happen when you split divergent series.
On the left side you have the divergent series 1 + 1/3 + 1/5 + 1/7 ... Let A be the name of this series.
On the right side you have the divergent series -1/2 - 1/4 - 1/6 - 1/8 ... Let B be the name of this series.

Let's say you have 1,000,000 terms in A and 1,000,000 terms in B (just to simplify it).
When you split B into -½(1 + 1/3 + 1/5 + 1/7 ...) - ½(1/2 + 1/4 + 1/6 + 1/8 ...), there would be 500,000 terms in each of those series. We'll call them B_1 and B_2.

You can't add A and B_1 together because there are twice as many terms in A than in B_1.

But if you start with 1,000,000 terms in A and 2,000,000 terms in B, there would be 1,000,000 terms in each of the series A, B_1 and B_2, so you could put them together without changing anything. But then it would be ½ ln2 from the start:
A: 1 . . . . . . +1/3 . . . . . . . +1/5 . . . . . . . . +1/7 ...
B: . -1/2 -1/4 . . . . -1/6 -1/8 . . . . -1/10 -1/12 . . . etc.
(Not 100% exactly ½ ln2, because we're using a finite number of terms.)

This is what would happen if we're letting the number of terms increase to infinity:
If the amount of terms in B increases twice as fast as in A, you get ½ ln2.
If the amount of terms in B increases with the same speed as in A, you get ln2.