Proof of inequality using Maclaurin series

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I want to prove that
$$
left|cos(x^3)-(1-fracx^62!+fracx^124!)right|leq frac6!
$$
for all $xinmathbbR$.




We know that
$$
cos x=1+sum_n=1^infty (-1)^nfracx^2n(2n)!,
$$
so
$$
cos(x^3)=1+sum_n=1^infty (-1)^nfracx^6n(2n)!
$$
for all $xinmathbbR$. Now we just have to prove that
$$
left|sum_n=3^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
or that
$$
left|-fracx^186!+sum_n=4^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
for all $xinmathbbR$. How can I prove that?




inequality power-series




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  • You cannot prove it, because the inequality holds in the other direction, at least in a nbd of zero.
    – uniquesolution
    Aug 22 at 9:54















up vote
0
down vote

favorite













I want to prove that
$$
left|cos(x^3)-(1-fracx^62!+fracx^124!)right|leq frac6!
$$
for all $xinmathbbR$.




We know that
$$
cos x=1+sum_n=1^infty (-1)^nfracx^2n(2n)!,
$$
so
$$
cos(x^3)=1+sum_n=1^infty (-1)^nfracx^6n(2n)!
$$
for all $xinmathbbR$. Now we just have to prove that
$$
left|sum_n=3^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
or that
$$
left|-fracx^186!+sum_n=4^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
for all $xinmathbbR$. How can I prove that?




inequality power-series




share|cite|improve this question






















  • You cannot prove it, because the inequality holds in the other direction, at least in a nbd of zero.
    – uniquesolution
    Aug 22 at 9:54













up vote
0
down vote

favorite









up vote
0
down vote

favorite












I want to prove that
$$
left|cos(x^3)-(1-fracx^62!+fracx^124!)right|leq frac6!
$$
for all $xinmathbbR$.




We know that
$$
cos x=1+sum_n=1^infty (-1)^nfracx^2n(2n)!,
$$
so
$$
cos(x^3)=1+sum_n=1^infty (-1)^nfracx^6n(2n)!
$$
for all $xinmathbbR$. Now we just have to prove that
$$
left|sum_n=3^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
or that
$$
left|-fracx^186!+sum_n=4^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
for all $xinmathbbR$. How can I prove that?




inequality power-series




share|cite|improve this question















I want to prove that
$$
left|cos(x^3)-(1-fracx^62!+fracx^124!)right|leq frac6!
$$
for all $xinmathbbR$.




We know that
$$
cos x=1+sum_n=1^infty (-1)^nfracx^2n(2n)!,
$$
so
$$
cos(x^3)=1+sum_n=1^infty (-1)^nfracx^6n(2n)!
$$
for all $xinmathbbR$. Now we just have to prove that
$$
left|sum_n=3^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
or that
$$
left|-fracx^186!+sum_n=4^infty (-1)^nfracx^6n(2n)!right|leqfrac6!
$$
for all $xinmathbbR$. How can I prove that?





inequality power-series





share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Aug 22 at 10:11









Bernard

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asked Aug 22 at 9:31









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  • You cannot prove it, because the inequality holds in the other direction, at least in a nbd of zero.
    – uniquesolution
    Aug 22 at 9:54

















  • You cannot prove it, because the inequality holds in the other direction, at least in a nbd of zero.
    – uniquesolution
    Aug 22 at 9:54
















You cannot prove it, because the inequality holds in the other direction, at least in a nbd of zero.
– uniquesolution
Aug 22 at 9:54





You cannot prove it, because the inequality holds in the other direction, at least in a nbd of zero.
– uniquesolution
Aug 22 at 9:54
















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