Math Problem Statement
Laske seuraavien jonojen raja-arvot mikäli mahdollista. (a) { (n^2 - 4) / (n + 5) }, (b) { (-1)^n (2n / (n^2 - 6)) }
Solution
Let’s solve the given limits.
Part (a)
The sequence is:
To find the limit as , divide the numerator and denominator by :
= \frac{n(1 - \frac{4}{n^2})}{1 + \frac{5}{n}}$$ As $$n \to \infty$$: - $$\frac{4}{n^2} \to 0$$ - $$\frac{5}{n} \to 0$$ Thus: $$\lim_{n \to \infty} a_n = \frac{n}{1} = n$$ --- For part (b): \### Part (b) The sequence is: $$b_n = (-1)^n \frac{2n}{n^2 - 6}.$$ To analyze its limit as $$n \to \infty$$: 1. Divide both the numerator and denominator by $$n^2$$: $$b_n = (-1)^n \frac{2n}{n^2 - 6} = (-1)^n \frac{\frac{2}{n}}{1 - \frac{6}{n^2}}.$$ 2. As $$n \to \infty$$: - $$\frac{2}{n} \to 0$$, - $$\frac{6}{n^2} \to 0$$. Thus: $$b_n \to 0 \text{ as } n \to \infty.$$ ### Final Results: 1. For part (a): The sequence grows without bound ($$\infty$$). 2. For part (b): The limit is $$0$$. Would you like a detailed explanation of any step or further elaboration on these calculations? Here are 5 related questions: 1. How do you determine whether a sequence diverges to infinity or oscillates? 2. Why is dividing by the highest power of $$n$$ helpful in limit calculations? 3. What would happen if the numerator in (a) had a smaller degree than the denominator? 4. Can part (b) converge if we remove the $$(-1)^n$$? 5. How does $$(-1)^n$$ affect the convergence properties of sequences? **Tip:** For alternating sequences like $$(-1)^n a_n$$, check if the absolute value $$|a_n|$$ converges first, then evaluate the oscillation.Ask a new question for Free
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Math Problem Analysis
Mathematical Concepts
Sequences and Limits
Rational Functions
Alternating Sequences
Formulas
lim_{n → ∞} a_n = Leading Term in Numerator / Leading Term in Denominator (for rational sequences)
Behavior of (-1)^n in alternating sequences
Theorems
Limits of Rational Functions
Behavior of Oscillating Sequences
Suitable Grade Level
Grades 10-12