Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

how to prove that this sequence not converges uniformly in all the real line? I don´t know how to do this problem :S , in fact I think that it converges uniformly in "all the real line" The sequence is $$ S_n \left( x \right) = \sum\limits_{k = 1}^n {\frac{{x^k }} {{k!}}} $$

share|cite|improve this question

The limit for $n\to\infty$ is easily seen to be $e^x-1$. However since $S_n(x)$ is a polynomial, it will always take on absolutely-large values for sufficiently large negative $x$. But for all such $x$, $e^x-1$ is close to $-1$.

Even without knowing what the limit is, however, the $S_n$'s are polynomials of alternating odd and even degrees. Therefore for any $N$ there is an $x$ that is so negative that $S_N(x)$ has a different sign from $S_{N+1}(x)$ and both are large.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

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