Help me understanding the harmonic analysis

I am not good in maths and got stuck on harmonic analysis please help me with it.

Hi Rahul,

Happy to help. Could you tell me which part of the analysis are you having trouble with?

Could you please explain how below equation is deduced in harmonic series of K groups
1 + 2 + … + 2^(k - 1) <= N

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I also didn’t get this. @Yash_Kumar sir can you please elaborate on this?