Explain this magic!?!?

Hello everyone!!

While researching on a series, I found something magical. Can someone explain this??

Let $S=1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+ ……$

$S= (1+\frac{1}{3}+\frac{1}{5}+……)+\frac{1}{2} [1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4}+……]$

$S=(1+\frac{1}{3}+\frac{1}{5}+....)+\frac{S}{2}$

$\frac{S}{2}=1+\frac{1}{3}+\frac{1}{5}+…$ $eq^{n} (i)$

Now from definition of S, $\frac{S}{2}=\frac{1}{2}+\frac{1}{4}+\frac{1}{6}+….$ $eq^{n} (ii)$

Comparing $$eq^{n} (i)$$ and $$eq^{n} (ii)$$ and transposing, We get $1-\frac{1}{2}+\frac{1}{3}-\frac{1}{4}+…..=0$

But obviously, $(1-\frac{1}{2})+(\frac{1}{3}-\frac{1}{4})+….>0$

Note by Pranjal Jain
3 years, 8 months ago

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Here S is not a converging series. The sum of the given series diverges. So you are treating S as a number, but its sum goes to infinity ( and infinity can not be considered as number , because it does not follow the properties of numbers). So unknowingly you are applying algebraic operations on infinity (for you it is S) , which is a flaw in this your something magical. !!!!!!!!

- 3 years, 8 months ago

Thanx dude! I got what you are saying! But I dont know much about convergence of a series! Any good and reliable source? Well, I tried to learn convergence from "Hall and Knight". I need some more help!

- 3 years, 8 months ago

Go for some UnderGraduate Maths book about sequence and series. You will find the complete conceptual information about the convergence and divergence. Hopefully, it will help you.

- 3 years, 8 months ago