The goal of this step-by-step walkthrough is to determine whether the series ∑n=1[infinity]en1+sin(n)\ converges or diverges.
Step 1: Understand the problem
The given series is an infinite summation that starts from n=1 to infinity. The nth term of the sequence is (e * n) / (1 + sin(n)). We need to ascertain if the sum of this series converges to a finite value or diverges to infinity.
Step 2: Explanation of the series
The series ∑n=1[infinity]en1+sin(n)\ is somewhat complex. The (e * n) part continuously grows as n increases, while the denominator, (1 + sin(n)), oscillates between 0 and 2.
Step 3: Consideration about the series’ behavior
Since the numerator continuously grows as n increases and the denominator only oscillates between static values, intuitively speaking, the overall value of the series would go to infinity as n tends towards infinity.
Step 4: Conclusion
In terms of mathematical conclusion, we say that the series ∑n=1[infinity]en1+sin(n)\ diverges. This essentially means that summing the series terms from n=1 to infinity does not result in a fixed, finite value. Instead, the sum goes to infinity.