$$\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n}<1$$ How can I show this holds? I have tried adding couple of terms in photomath and it seems to hold, cannot prove it tho
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1Do you know sum of terms of G.P. ? – Get_ Maths Sep 24 '22 at 16:48
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See for example https://math.stackexchange.com/q/1448626/42969 or https://math.stackexchange.com/q/3837387/42969 – Martin R Sep 24 '22 at 16:52
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This is a geometric series... To compute it, let $S_n = (x^0+x^1+x^2+\ldots +x^n)$. Then compute $(1-x)S_n$ to get $(1-x)S_n=(1-x^{n+1})$ and finally replace $x$ with $1/2$ to conclude. – Surb Sep 24 '22 at 16:53
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You can prove this by induction. – CyclotomicField Sep 24 '22 at 16:54
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4Does this answer your question? Prove by induction $\sum \frac {1}{2^n} <1$ – Dietrich Burde Sep 24 '22 at 16:56
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Since there is no $n$ on the right side, there is no way to induct.
What you can do is show by induction that $\frac{1}{2}+\frac{1}{4}+\cdots+\frac{1}{2^n} =1-\frac1{2^n} $ from which the conclusion immediately follows.
marty cohen
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