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Prove that $\exists\,! \,\lambda \in (1/5,1/4)$ such that $\displaystyle\frac{1}{2\pi}\int_0^{2\pi}e^{\sin x}\,\mathrm{d}x=e^{\lambda}$

Now the integral can be written as

$$ \frac{1}{2\pi}\int_0^{2\pi}e^{\sin x}\,\mathrm{d}x = \int_0^{1}e^{\sin 2\pi x}\,\mathrm{d}x = \sum_{k\geq 0} \int_0^{1} \frac{\sin^k(2\pi x)}{k!} \,\mathrm{d}x = \sum_{k\geq 0} \frac{1}{4^k(k!)^2} $$

Which is the same as $J_0(1)$, where $J$ is the bessel function of the first kind. Now one can also rewrite the problem by using $e^x = \sum_{k=0} x^k/k!$. So

\begin{array}{ccc} e^{1/5} & < & \int_0^{1}e^{\sin 2\pi x}\,\mathrm{d}x & < & e^{1/4}\\ \sum_{k\geq0} \frac{1}{5^k k!} & < & \sum_{k\geq 0} \frac{1}{4^k(k!)^2} & < & \sum_{k\geq0} \frac{1}{4^k k!} \end{array}

Alas the latter sums seems just as hard to prove than the integral form. Any help or suggestions would be appreciated.

  • $\begingroup$ Jordan's inequality worked to show $\lambda \in (0.1,0.4)$ $\endgroup$ – John Fernley Jun 26 '14 at 18:30
  • $\begingroup$ One inequality at least is obvious since $1\leq k!, \ \forall \ k \geq 0 \iff 4^kk! \leq 4^k(k!)^2, \ \forall \ k \geq 0 \iff \sum_{k\geq 0}\frac{1}{4^k(k!)^2}\leq \sum_{k\geq 0}\frac{1}{4^kk}$ $\endgroup$ – Surb Jun 26 '14 at 18:44
  • $\begingroup$ The other inequality looks to largely come down to the $k=1$ term wolframalpha.com/input/?i=k%21+4%5Ek+-+5%5Ek $\endgroup$ – John Fernley Jun 26 '14 at 19:10

Anticlimax ahead:

$$\sum_{k=0}^\infty \frac{1}{4^k(k!)^2} < \sum_{k=0}^\infty \frac{1}{4^k\,k!}$$

follows since $4^k(k!)^2 \geqslant 4^k\, k!$ for all $k$ and the inequality is strict for $k\geqslant 2$.

For the inequality

$$\sum_{k=0}^\infty \frac{1}{5^k\,k!} < \sum_{k=0}^\infty \frac{1}{4^k(k!)^2},$$

we note that the terms for $k = 0$ are equal in both series, so it suffices to see

$$\sum_{k=2}^\infty \frac{1}{k!}\left(\frac{1}{5^k} - \frac{1}{4^k\,k!}\right) \leqslant \sum_{k=2}^\infty \frac{1}{5^k\,k!} \leqslant \frac{1}{50}\sum_{\nu=0}^\infty \frac{1}{15^\nu} = \frac{15}{14\cdot 50} = \frac{3}{140} < \frac{1}{20} = \frac{1}{4^1(1!)^2} - \frac{1}{5^1\,1!}.$$


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