Prove by Mean Value Theorem $frac{x}{1+x}<ln(1+x)0$











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Prove for $x>0$
$$
frac{x}{1+x}<ln(1+x)<x
$$
I tried writing $ln(1+x)=ln(1+x)-ln(1)$ and using the MVT for the $(1,1+x)$ interval. I eventually could prove the inequality but how do I have to prove even for $(0,1)$










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  • can it be done without Taylor?
    – user224677
    Mar 18 '15 at 18:08










  • I know but we haven't quite studied Taylor series so if you could help with some hint, I just don't know how to prove for (0;1) interval because the rest I have done
    – user224677
    Mar 18 '15 at 18:11










  • I will think on it; I don't currently have it solved either :)
    – graydad
    Mar 18 '15 at 18:13















up vote
3
down vote

favorite
1












Prove for $x>0$
$$
frac{x}{1+x}<ln(1+x)<x
$$
I tried writing $ln(1+x)=ln(1+x)-ln(1)$ and using the MVT for the $(1,1+x)$ interval. I eventually could prove the inequality but how do I have to prove even for $(0,1)$










share|cite|improve this question
























  • can it be done without Taylor?
    – user224677
    Mar 18 '15 at 18:08










  • I know but we haven't quite studied Taylor series so if you could help with some hint, I just don't know how to prove for (0;1) interval because the rest I have done
    – user224677
    Mar 18 '15 at 18:11










  • I will think on it; I don't currently have it solved either :)
    – graydad
    Mar 18 '15 at 18:13













up vote
3
down vote

favorite
1









up vote
3
down vote

favorite
1






1





Prove for $x>0$
$$
frac{x}{1+x}<ln(1+x)<x
$$
I tried writing $ln(1+x)=ln(1+x)-ln(1)$ and using the MVT for the $(1,1+x)$ interval. I eventually could prove the inequality but how do I have to prove even for $(0,1)$










share|cite|improve this question















Prove for $x>0$
$$
frac{x}{1+x}<ln(1+x)<x
$$
I tried writing $ln(1+x)=ln(1+x)-ln(1)$ and using the MVT for the $(1,1+x)$ interval. I eventually could prove the inequality but how do I have to prove even for $(0,1)$







calculus inequality logarithms






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edited Nov 18 at 6:35









Martin Sleziak

44.5k7115268




44.5k7115268










asked Mar 18 '15 at 17:55









user224677

161




161












  • can it be done without Taylor?
    – user224677
    Mar 18 '15 at 18:08










  • I know but we haven't quite studied Taylor series so if you could help with some hint, I just don't know how to prove for (0;1) interval because the rest I have done
    – user224677
    Mar 18 '15 at 18:11










  • I will think on it; I don't currently have it solved either :)
    – graydad
    Mar 18 '15 at 18:13


















  • can it be done without Taylor?
    – user224677
    Mar 18 '15 at 18:08










  • I know but we haven't quite studied Taylor series so if you could help with some hint, I just don't know how to prove for (0;1) interval because the rest I have done
    – user224677
    Mar 18 '15 at 18:11










  • I will think on it; I don't currently have it solved either :)
    – graydad
    Mar 18 '15 at 18:13
















can it be done without Taylor?
– user224677
Mar 18 '15 at 18:08




can it be done without Taylor?
– user224677
Mar 18 '15 at 18:08












I know but we haven't quite studied Taylor series so if you could help with some hint, I just don't know how to prove for (0;1) interval because the rest I have done
– user224677
Mar 18 '15 at 18:11




I know but we haven't quite studied Taylor series so if you could help with some hint, I just don't know how to prove for (0;1) interval because the rest I have done
– user224677
Mar 18 '15 at 18:11












I will think on it; I don't currently have it solved either :)
– graydad
Mar 18 '15 at 18:13




I will think on it; I don't currently have it solved either :)
– graydad
Mar 18 '15 at 18:13










1 Answer
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By the mean value theorem, given $x > 0$, there exists $c in (0,x)$ such that $f(x) - f(0) = f'(c)x$, i.e., $ln(1 + x) = frac{x}{1 + c}$. Since $0 < c < x$, $frac{1}{1 + x} < frac{1}{1 + c} < 1$. Therefore $frac{x}{1 + x} < frac{x}{1 + c} < x$, i.e.,



$$frac{x}{1 + x} < ln(1 + x) < x.$$






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    By the mean value theorem, given $x > 0$, there exists $c in (0,x)$ such that $f(x) - f(0) = f'(c)x$, i.e., $ln(1 + x) = frac{x}{1 + c}$. Since $0 < c < x$, $frac{1}{1 + x} < frac{1}{1 + c} < 1$. Therefore $frac{x}{1 + x} < frac{x}{1 + c} < x$, i.e.,



    $$frac{x}{1 + x} < ln(1 + x) < x.$$






    share|cite|improve this answer

























      up vote
      2
      down vote













      By the mean value theorem, given $x > 0$, there exists $c in (0,x)$ such that $f(x) - f(0) = f'(c)x$, i.e., $ln(1 + x) = frac{x}{1 + c}$. Since $0 < c < x$, $frac{1}{1 + x} < frac{1}{1 + c} < 1$. Therefore $frac{x}{1 + x} < frac{x}{1 + c} < x$, i.e.,



      $$frac{x}{1 + x} < ln(1 + x) < x.$$






      share|cite|improve this answer























        up vote
        2
        down vote










        up vote
        2
        down vote









        By the mean value theorem, given $x > 0$, there exists $c in (0,x)$ such that $f(x) - f(0) = f'(c)x$, i.e., $ln(1 + x) = frac{x}{1 + c}$. Since $0 < c < x$, $frac{1}{1 + x} < frac{1}{1 + c} < 1$. Therefore $frac{x}{1 + x} < frac{x}{1 + c} < x$, i.e.,



        $$frac{x}{1 + x} < ln(1 + x) < x.$$






        share|cite|improve this answer












        By the mean value theorem, given $x > 0$, there exists $c in (0,x)$ such that $f(x) - f(0) = f'(c)x$, i.e., $ln(1 + x) = frac{x}{1 + c}$. Since $0 < c < x$, $frac{1}{1 + x} < frac{1}{1 + c} < 1$. Therefore $frac{x}{1 + x} < frac{x}{1 + c} < x$, i.e.,



        $$frac{x}{1 + x} < ln(1 + x) < x.$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Mar 18 '15 at 18:15









        kobe

        34.5k22247




        34.5k22247






























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