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Math  /  Calculus

Question1. The sequence an=3n2+11−n3a_{n}=\frac{3 n^{2}+1}{1-n^{3}} (A) converges to 0 . (B) converges to -1 . (C) converges to -2 . (D) is divergent.
2. The sequence an=(−1)n5n3n72na_{n}=(-1)^{n} \frac{5^{n} 3^{n}}{7^{2 n}} (A) converges to 1 . (B) converges to 0 . (C) converges to 2 . (D) is divergent.
3. The sequence an=3n+1n+1a_{n}=\frac{3^{n}+1}{n+1} (A) converges to 1 . (B) converges to 0 . (C) converges to 2 . (D) is divergent.

Studdy Solution
Consider the sequence an=3n+1n+1 a_{n}=\frac{3^{n}+1}{n+1} .
As n→∞ n \to \infty , the dominant term in the numerator is 3n 3^n and in the denominator is n n .
an≈3nn a_{n} \approx \frac{3^n}{n}
Since 3n 3^n grows exponentially and n n grows linearly, 3nn→∞ \frac{3^n}{n} \to \infty .
Therefore, the sequence is divergent.
The answers are:
1. (A) converges to 0.
2. (B) converges to 0.
3. (D) is divergent.

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