The notation ∑n=1∞ (-1)^n/n represents an infinite series, where n starts at 1 and increases up to infinity. The term (-1)^n/n means that the sign alternates between positive and negative, with the value of n being divided by itself.
To find the sum of an infinite series, you can use a specific formula or method depending on the type of series. For example, in this series, you can use the Alternating Series Test or the Ratio Test to determine if the series converges or diverges. If it converges, you can then use the formula for the sum of an alternating series to find the value of the sum.
The Alternating Series Test is a method used to determine if an alternating series, such as the one in this problem, converges or diverges. It states that if the series alternates in sign and the terms decrease in absolute value, then the series converges. However, this test does not tell us the value of the sum, only if it converges or diverges.
To use the Ratio Test, you take the limit of the absolute value of the ratio of the (n+1)th term to the nth term as n approaches infinity. If the limit is less than 1, then the series converges. If the limit is greater than 1, then the series diverges. And if the limit is equal to 1, then the test is inconclusive, and you would need to use another method to determine convergence.
To solve this series, we would first use the Alternating Series Test to determine that it converges. Then, we would use the formula for the sum of an alternating series, which is the limit as n approaches infinity of the partial sum of the series. In this case, the partial sum would be (1-1/2)+(1/3-1/4)+(1/5-1/6)+...+(1/n-1/(n+1)). This limit can be simplified to 1-1/(n+1), which as n approaches infinity, becomes 1. Therefore, the sum of this infinite series is 1.