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Answer 1 · 2017-02-16T17:10:14

The series:

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

is not convergent.

Consider the function:

#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.