Question #ac3ad

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Answer 1 · 2017-02-16T17:10:14

The series:

$\infty \sum n = 1 {(- 1)}^{n} sec (\frac{1}{n^{3}})$ $sum_(n=1)^oo (-1)^nsec(1/n^3)$

is not convergent.

Consider the function:

$f (x) = sec (\frac{1}{x^{3}}) = \frac{1}{cos (\frac{1}{x^{3}})}$ $f(x) = sec(1/x^3) = 1/cos(1/x^3)$

we have:

$lim_(x->oo)sec(1/x^3) = lim_(y->0^+) 1/cos(y) = 1$

and so:

$lim_(n->oo) sec(1/n^3) = 1$

and:

$lim_(n->oo) (-1)^nsec(1/n^3)$

is indeterminate.

Thus the series:

$sum_(n=1)^oo (-1)^nsec(1/n^3)$

is not satisfying Cauchy's necessary condition for convergence:

$lim_(n->oo) a_n = 0$

and cannot be convergent.