Question

summation from n=1 to infinity [(5^n+1)/(6^n+n)]
converge or diverge?

Answer 1

\[\sum_{n=1}^{\infty} (5^{n}+1)/(6^{n}+n)\]

Answer 2

converge.\[\text{Limit}\left[\frac{1+5^n}{6^n+n},n\to \text{Infinity}\right]=0 \]Notice that the denominator becomes much larger than the numerator as x increases.

Answer 3

what type of test can we use?

Answer 4

@robtobey it does not guarantee limit is 0 or converges so series is converges

for example 1/n n=1 to infinite its limit is 0 but it is diverges..