## experimentX 3 years ago Math.

1. experimentX

it is given to prove that x/(1+x) < ln(1+x) < x in the interval [1,1+t] t>0

2. experimentX

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4. experimentX

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5. Callisto

But then, you still haven't considered lnx...

6. experimentX

why do you need to prove ... ln(2) > 1/2 ?

7. Callisto

It's not prove.. but.. we have to consider lnx..

8. experimentX

why??

9. experimentX

By considering ln x in [1; 1 + t] (t > 0), ln x in [1; 1 + t] (t > 0), minimum bound : ln(x) = 1 => x = e

10. experimentX

x/(1+x) < ln(1+x) < x ... perhaps we can use the same argument as we used above.

11. experimentX

$(x+1)^2 > 1+x > 1 \text{ for all x > 0} \\ {1 \over (1+x)^2} < {1 \over 1 +x} < 1 \text{ for all x > 0} \\ \int_0^x {1 \over (1+x)^2} \;dx < \int_0^x {1 \over 1 +x}\;dx < \int_0^x1 \; dx \\ {x \over 1+x} < \ln(1+x) < x$

12. experimentX

$\ln (x) \text{ in } [1, 1+t], t>0$ perhaps this would mean show that $$\ln(1)<{t \over 1 +t} < \ln(1+t)$$

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20. Callisto

f(b) - f(a) / (b-a) ?

21. experimentX

yep!!

22. experimentX

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23. Callisto

OMG! My super smart brother!!

24. experimentX

lol ... not that smart. Just copied half from your prev answer.