27: There is just one trouble, at zero. There is only one obvious inequality, the given function can be estimated from below by ignoring the 1 in the denominator. Unfortunately, the resulting test function gives a convergent integral (the test function is continuous on [0,1]), so no conclusion is possible.

The only hope is now the Limit Comparison test. Since the exponential is rather annoying, one can ask: How does it look like when x is close to 0? From the Taylor series we know that for x near zero,

e ^x ∼ 1 + x + x²/2+...

and therefore

e ^x2 ∼ 1 + x² + x⁴/2+...

Use this to set up a limit comparison around zero.

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