Answer to Question #308870 in Real Analysis for Dhruv rawat

Question

Test whether the series ∞Σn=0 1/(n^5+x^3) converges uniformly or not

Answer 1

Alternative method

We consider {a_n} = \frac{1}{{{n^5} + {x^3}}}\and {b_n} = \frac{1}{{{n^5}}}\     

Here we can see both the series \sum {a_n}\and $\sum {b_n}$ converge. 

Now

\begin{array}{l} \mathop {\lim }\limits_{n \to \infty } \frac{{{a_n}}}{{{b_n}}}\\ = \mathop {\lim }\limits_{n \to \infty } \frac{{\frac{1}{{{n^5} + {x^3}}}}}{{\frac{1}{{{n^5}}}}}\\ = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^5}}}{{{n^5} + {x^3}}}\\ = \mathop {\lim }\limits_{n \to \infty } \frac{{{n^5}}}{{{n^5}\left( {1 + \frac{{{x^3}}}{{{n^5}}}} \right)}}\\ = \mathop {\lim }\limits_{n \to \infty } \frac{1}{{1 + \frac{{{x^3}}}{{{n^5}}}}}\\ = \frac{1}{{1 + \frac{{{x^3}}}{{{\infty ^5}}}}} = \frac{1}{{1 + 0}} = 1 \end{array}\

Therefore, by limit comparison test the series \sum\limits_{n = 0}^\infty {{a_n}} = \sum\limits_{n = 0}^\infty {\frac{1}{{{n^5} + {x^3}}}} \ converges.