Limit Comparison Test for Convergence of an Infinite Series - Page 5

Questions
  • How do you proof lim x → ∞ (n^(x+1)) / ((x+1)!) = 0 ?
  • Lim x→3 [8x/ (x−3)^2 ] Why does the result is infinity positive?
  • Is the series #\sum_(n=0)^\infty n^n/(4^n n!)# absolutely convergent, conditionally convergent or divergent?
  • What is #lim_(n->oo) 1/n(sin(pi/n) + sin((2pi)/n) + ... + sin((2npi)/n))# ?
  • If I want to test the series #sum_(n=1)^oo(n^2+2^n)/(1-e^(n+1))# for convergence, what would be the best test to use and why?
  • How to evaluate the limit #lim_(n -> oo) sum_(j=1)^n(5j)/n^2#?
  • What's the difference between: undefined, does not exist and infinity?
  • What is #lim_(x->oo) (sqrt(xsqrt(x+sqrt(x)))-sqrt(x))# ?
  • What is the limit as #n# approaches infinity of #sqrt(n^2 + n) - n#?
  • What is #lim x->∞# of #(7x+1)/(11x-6)#?
  • What is #lim x-> ∞# of #(x^4+3)/(2x^2-5)#?
  • What is the #lim (sqrt(x^2+3x)-x)#, as x approaches infinity?
  • What is #\lim _ { n \rightarrow \infty } \sqrt { x } - x#?
  • What is the limit of (1+(4/x))^x as x approaches infinity?
  • Prove that #limxlnx# = 0 as x approaches positive 0, given that #lim(lnx)/x# = 0 as x approaches positive infinity?
  • Evaluate #lim_(n->oo) 1/n^4 prod_(j=1)^(2n) (n^2+j^2)^(1/n)#?
  • How do you solve #lim_(n->oo) (3^n + 5^n + 7^n)^(1/n) = #?
  • Lim x → ∞{(x+a)(x+b)}^1/2 -x) equals?
  • If #sum_k a_n = sum_k (-2n^4 + 5n^2 - 3)/(n^11 - 9n^3 - 9)# and #sum_k b_n = sum_k 1/n^beta#, then for what value of #beta# could allow #a_n/b_n# to converge?
  • How do you evaluate #\lim _ { x \rightarrow + \infty } \frac { 2} { \sqrt { x + 1} }#?