Question

Determine whether the following series converges or diverges. Say what comparison test you used and what series you used for comparison: Σ_{n=1}[infinity] {e^{n}+1}{n e^{n}+1}

Answer 1

The series Σ_{n=1}^∞ (e^n+1)(ne^n+1) diverges.

Step-by-step explanation:

Comparison Test

To determine if the series Σ_{n=1}^∞ (e^n+1)(ne^n+1) converges or diverges, we can use the comparison test.

Let's compare it with the series Σ_{n=1}^∞ ne^n.

Comparison

For n ≥ 1, we have (e^n+1)(ne^n+1) ≥ ne^n.

This is because e^n+1 and e^n+1 are both greater than 1 for n ≥ 1.

Therefore, since Σ_{n=1}^∞ ne^n diverges by the ratio test, the given series Σ_{n=1}^∞ (e^n+1)(ne^n+1) also diverges.