Determining the Convergence of the Series: ∑₁^∞ (-1)ⁿn⁶/7ⁿ

When dealing with infinite series, it is essential to determine whether the series converges or diverges. The convergence or divergence of a series can have significant implications in various areas of mathematics, including calculus, analysis, and numerical methods.

In this article, we will analyze the convergence of the following infinite series:

∑₁^∞ (-1)ⁿn⁶/7ⁿ

To determine the convergence or divergence of this series, we will use several series tests, including the alternating series test and the ratio test.

Alternating Series Test

The alternating series test is a useful tool for analyzing the convergence of series where the terms alternate in sign (i.e., positive and negative terms).

The alternating series test states that if a series satisfies the following conditions:

1. The terms of the series are alternating in sign (i.e., a₁, -a₂, a₃, -a₄, …).
2. The absolute value of the terms is decreasing (i.e., a₁ ≥ |a₂| ≥ |a₃| ≥ …).
3. The limit of the terms as n → ∞ is 0 (i.e., limₙ→∞ aₙ = 0).

Then the series converges.

Let's apply the alternating series test to the given series:

∑₁^∞ (-1)ⁿn⁶/7ⁿ

1. The terms of the series are alternating in sign, as (-1)ⁿ alternates between 1 and -1.
2. The absolute value of the terms is decreasing, as |n⁶/7ⁿ| is a decreasing function of n.
3. The limit of the terms as n → ∞ is 0, as limₙ→∞ n⁶/7ⁿ = 0.

Since all the conditions of the alternating series test are satisfied, we can conclude that the series converges.

Ratio Test

The ratio test is another useful tool for analyzing the convergence of series. The ratio test states that if the limit of the ratio of consecutive terms in a series is less than 1, then the series converges. If the limit is greater than 1, then the series diverges. If the limit is equal to 1, the test is inconclusive.

Let's apply the ratio test to the given series:

∑₁^∞ (-1)ⁿn⁶/7ⁿ

The ratio of consecutive terms is:

limₙ→∞ ((-1)ⁿ⁺¹(n+1)⁶/7ⁿ⁺¹) / ((-1)ⁿn⁶/7ⁿ) = limₙ→∞ (n+1)⁶/n⁶ × 1/7 = 1/7

Since the limit of the ratio is less than 1, the series converges.

Conclusion

By applying the alternating series test and the ratio test, we have determined that the series:

∑₁^∞ (-1)ⁿn⁶/7ⁿ

converges. The alternating series test confirms the convergence, as the series satisfies the required conditions. The ratio test further supports the convergence, as the limit of the ratio of consecutive terms is less than 1.

In summary, the given infinite series converges, and this result can be useful in various mathematical applications and analyses.