What makes an infinite series converge?

An infinite series converges if the sequence of its partial sums approaches a finite limit. In simpler terms, the sum of an infinite number of terms gets closer and closer to a specific, real number.

Understanding Convergence

To understand what makes a series converge, consider these key aspects:

• Partial Sums: For an infinite series ∑ a_n (where 'n' goes from 1 to infinity), the partial sums are defined as:

  • S₁ = a₁
  • S₂ = a₁ + a₂
  • S₃ = a₁ + a₂ + a₃
  • ...
  • S_k = a₁ + a₂ + ... + a_k

• The Limit of Partial Sums: An infinite series converges if the limit of the sequence of partial sums (S_k) exists and is a finite number as k approaches infinity. Mathematically, this is expressed as:

  lim_k→∞ S_k = L (where L is a finite number)

  If this limit exists, we say that the series converges to L, and L is the sum of the infinite series.

• Divergence: If the limit of the sequence of partial sums either does not exist (oscillates) or goes to infinity (positive or negative), then the infinite series diverges.

Necessary Conditions for Convergence

While not sufficient on their own, these are conditions that must be met for convergence:

• The nth Term Test (Divergence Test): If lim_n→∞ a_n ≠ 0, then the series ∑ a_n diverges. This test cannot prove convergence; it can only prove divergence. If lim_n→∞ a_n = 0, the series might converge, but further testing is required.

Convergence Tests

Many tests can determine if a series converges. Here are a few common ones:

• Geometric Series Test: A geometric series ∑ ar^n-1 converges if |r| < 1, where 'a' is the first term and 'r' is the common ratio. Its sum is a/(1-r).

• Integral Test: If f(x) is a continuous, positive, and decreasing function for x ≥ 1, and a_n = f(n), then the series ∑ a_n and the integral ∫₁^∞ f(x) dx either both converge or both diverge.

• Comparison Test: If 0 ≤ a_n ≤ b_n for all n, and ∑ b_n converges, then ∑ a_n also converges. Conversely, if a_n ≥ b_n ≥ 0 for all n, and ∑ b_n diverges, then ∑ a_n also diverges.

• Limit Comparison Test: If lim_n→∞ (a_n/b_n) = c, where 0 < c < ∞ (c is a finite, positive number), then ∑ a_n and ∑ b_n either both converge or both diverge.

• Ratio Test: Let L = lim_n→∞ |a_n+1/a_n|.

  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.

• Root Test: Let L = lim_n→∞ √n.

  • If L < 1, the series converges absolutely.
  • If L > 1, the series diverges.
  • If L = 1, the test is inconclusive.

• Alternating Series Test: If a series is alternating (terms alternate in sign), the absolute value of the terms is decreasing, and the limit of the terms is zero, then the alternating series converges.

Example

Consider the series ∑ (1/2)ⁿ (from n=1 to infinity). This is a geometric series with a = 1/2 and r = 1/2. Since |r| = |1/2| < 1, the series converges to (1/2) / (1 - 1/2) = (1/2) / (1/2) = 1.

In Summary

An infinite series converges when the sum of its terms approaches a finite limit as you add more and more terms. This convergence is determined by analyzing the behavior of its partial sums and using various convergence tests to assess its ultimate sum.