How to Find Sum of Alternating Series?

Answer: To find the sum of an alternating series, use the formula for the sum of an infinite alternating series: Sum = a / (1 – r), where “a” is the first term and “r” is the common ratio between consecutive terms.

The sum of an infinite alternating series can be found using a specific formula. If the alternating series is in the form:

a − ar + ar₂− ar₃+…

where a is the first term and r is the common ratio between consecutive terms, the sum S of the series is given by the formula:

S = [Tex]\frac{a}{1 – r}[/Tex]

This formula is derived from the concept of geometric series. It’s important to note that the formula is applicable only when the absolute value of the common ratio ∣r∣ is less than 1, ensuring that the series converges to a finite sum. If ∣r∣ is greater than or equal to 1, the series diverges, and the sum is undefined.

Solved Examples

Example 1: Consider the alternating series [Tex]\sum_{n=1}^{\infty} (-1)^{n+1} \frac{1}{n} [/Tex], Find the sum of first 5 terms.

Solution:

This series alternates signs and consists of terms \( \frac{1}{n} \).

To find the sum, you can approximate it by adding up terms.

Now, let’s find the sum of the first 5 terms:

[Tex]S_5 = \frac{1}{1} – \frac{1}{2} + \frac{1}{3} – \frac{1}{4} + \frac{1}{5}[/Tex]

[Tex]S_5 = 1 – 0.5 + \frac{1}{3} – 0.25 + 0.2 = 0.95[/Tex]

Example 2: Find the Sum of the Infinite G.P: 0.5, 1, 2, 4, 8, …

Solution: