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:
Formula for the Sum of Infinite G.P: ?1−?1−ra ; r≠0
a = 0.5, r = 2
S∞= (0.5)/(1-2) = 0.5/(-1)= -0.5
Contact Us