I can't find any mistake here

Let, S=112+1314+15+....S_\infty=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+....

By calculating, we find that:

S1=1S_1=1

S2=12S_2=\frac{1}{2}

S3=0.833333...S_3=0.833333...

S4=0.5833333...S_4=0.5833333...

S5=0.78333...S_5=0.78333...

S6=0.616666...S_6=0.616666...

S7=0.759523809...S_7=0.759523809...

S8=0.634523809...S_8=0.634523809...

S9=0.7445492...S_9=0.7445492...

S10=0.64563492...S_{10}=0.64563492...

We'll notice that the Sn0.7S_n\rightarrow0.7 as nn\rightarrow\infty and in fact, it is ln2=0.69314718...\ln 2=0.69314718...

Now, S=112+1314+15+....S_\infty=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+....

And 2S=21+2324+2526+2728+29210+...2S_\infty=2-1+\frac{2}{3}-\frac{2}{4}+\frac{2}{5}-\frac{2}{6}+\frac{2}{7}-\frac{2}{8}+\frac{2}{9}-\frac{2}{10}+...

=1+2312+2513+2714+2915+...=1+\frac{2}{3}-\frac{1}{2}+\frac{2}{5}-\frac{1}{3}+\frac{2}{7}-\frac{1}{4}+\frac{2}{9}-\frac{1}{5}+...

=112+(2313)14+(2515)...=1-\frac{1}{2}+(\frac{2}{3}-\frac{1}{3})-\frac{1}{4}+(\frac{2}{5}-\frac{1}{5})-...

=112+1314+15+...=1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+\frac{1}{5}+...

=S=S_\infty

So, 2S=S2S_\infty=S_\infty

We know that, SS_\infty is not equal to 00.

Then 2=12=1!!

Please help me solving the mistake.

Note by Fahim Muhtamim
2 weeks, 4 days ago

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1 vote

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Rearranging the terms of a sum that is not absolutely convergent may change the value of the sum. In fact, there is a wonderful result that if a series is convergent but not absolutely convergent, then for any real number r,r, you can rearrange the terms of the series so that it sums to r.r. Here is a nice reference.

Patrick Corn - 2 weeks ago

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Your mistake is when you put it as 112+(2313)..1-\frac{1}{2}+(\frac{2}{3}-\frac{1}{3})...

The gaps between 23and13,25and15,29and19\frac{2}{3} and \frac{1}{3}, \frac{2}{5}and \frac{1}{5},\frac{2}{9} and \frac{1}{9}. Increase each time. With this logic you can prove that there are twice as many even numbers as odd numbers.

1,3,2,5,7,4,9,11,6..1,3,2,5,7,4,9,11,6..

Chris Sapiano - 2 weeks, 1 day ago

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