Question

Arată ca 1/2 + 1/4 + 1/8 +...+ 1/1024 < 1
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Answer 1

Răspuns:

Explicație pas cu pas:

$S=\dfrac{1}{2}+ \dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{512}+\dfrac{1}{1024}\\2*S=1+\dfrac{1}{2}+ \dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{512}\\2*S-S=S=1+\dfrac{1}{2}+ \dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{512}-(\dfrac{1}{2}+ \dfrac{1}{4}+\dfrac{1}{8}+...+\dfrac{1}{512}+\dfrac{1}{1024})=1-\dfrac{1}{1024}=\dfrac{1024}{1024}-\dfrac{1}{1024}=\dfrac{1023}{1024} <1$