2
$\begingroup$

Consider the series sigma (from k=1 to infinity) (-1)^k.(x+k/k^2) for x within [0,1].

I deduce that the series (-1)^k.(x/k^2) is uniformly convergent from using the M-test for convergence, and I assume that the series (-1)^k.(1/k) is not uniformly convergent because it has no x term so isn't a series of functions satisfying the definition of uniform convergence of series. (I know that it is conditionally convergent, however.)

Thus is the original series uniformly convergent?

$\endgroup$
1
  • $\begingroup$ Dear me I totally meant uniformly convergent, sorry! Just edited it. $\endgroup$ Oct 14 '18 at 19:07
0
$\begingroup$

Hint. One has $$ \sup_{x \in [0,1]}\left|(-1)^k\frac{x+k}{k^2} \right|=\frac1{k^2}+\frac1k\qquad (k\ge1) $$ thus the given series does not converge uniformly over $[0,1]$.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.