The point of convergence of an infinite series is the value that the sum of the series approaches as the number of terms increases infinitely. You can find it using various methods, depending on the type of series.
1. Geometric Series:
A geometric series has a common ratio r between consecutive terms.
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Formula: The sum of an infinite geometric series converges if the absolute value of the common ratio, |r|, is less than 1. The point of convergence is given by:
- S = a / (1 - r), where a is the first term.
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Example: The series 1 + 1/2 + 1/4 + 1/8 + ... has a common ratio r = 1/2. Since |r| = 1/2 < 1, the series converges. The point of convergence is:
- S = 1 / (1 - 1/2) = 2.
2. Ratio Test:
The ratio test is a general method used to determine if a series converges or diverges.
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Formula: Calculate the limit as n approaches infinity of the absolute value of the ratio of the (n+1)th term to the nth term:
- L = lim (n→∞) |a<sub>n+1</sub> / a<sub>n</sub>|
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Conditions:
- If L < 1, the series converges absolutely.
- If L > 1, the series diverges.
- If L = 1, the test is inconclusive.
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Example: For the series Σ(n=1 to ∞) n/2<sup>n</sup>, the ratio test gives:
- L = lim (n→∞) |(n+1)/2<sup>n+1</sup> * 2<sup>n</sup>/n| = 1/2.
- Since L < 1, the series converges.
3. Integral Test:
The integral test can be used for series with positive terms that are decreasing functions.
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Formula: If f(x) is a continuous, positive, and decreasing function on the interval [1, ∞), then the series Σ(n=1 to ∞) f(n) converges if and only if the improper integral ∫(1 to ∞) f(x) dx converges.
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Example: For the series Σ(n=1 to ∞) 1/n<sup>2</sup>, the integral test gives:
- ∫(1 to ∞) 1/x<sup>2</sup> dx = [ -1/x ]<sub>1</sub><sup>∞</sup> = 1.
- Since the integral converges, the series also converges.
4. Comparison Tests:
Comparison tests compare the given series to a known convergent or divergent series.
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Direct Comparison Test: If 0 ≤ a<sub>n</sub> ≤ b<sub>n</sub> for all n and Σ(n=1 to ∞) b<sub>n</sub> converges, then Σ(n=1 to ∞) a<sub>n</sub> also converges.
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Limit Comparison Test: If lim (n→∞) a<sub>n</sub> / b<sub>n</sub> = c, where c is a finite positive number, then Σ(n=1 to ∞) a<sub>n</sub> and Σ(n=1 to ∞) b<sub>n</sub> either both converge or both diverge.
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Example: For the series Σ(n=1 to ∞) 1/(n<sup>2</sup> + 1), we can compare it to the series Σ(n=1 to ∞) 1/n<sup>2</sup>, which we know converges. Using the direct comparison test, we find that 1/(n<sup>2</sup> + 1) ≤ 1/n<sup>2</sup> for all n. Therefore, the series Σ(n=1 to ∞) 1/(n<sup>2</sup> + 1) also converges.
5. Other Tests:
There are other tests, such as the alternating series test, root test, and p-series test, which can be used to determine the convergence of specific types of series.
Conclusion:
Finding the point of convergence of an infinite series involves understanding the behavior of the series as the number of terms increases infinitely. This can be achieved by employing various tests and methods, each suitable for different types of series. By applying these tests, you can determine if the series converges and, if so, identify its point of convergence.
